text: 2 of a short-lived rotating and nonlinear charged black exponentially, being this last the actual chiral plasma in- hole (RNCBH) [15, 16], that is, a black hole described stability. In our understanding, this ﬁeld increase would by a theory of the electromagnetism more general than suggest that NLED might be at action inside just-born Maxwell’s. This might happen because electromagnetic pulsars. For further details on magnetic ﬁeld ampliﬁca- ﬁelds could easily surpass certain scale ﬁelds (intrin- tion, see Ref. [40]. sically associated with nonlinear theories of the elec- Meanwhile, the gravitational eﬀects on the neutrino os- −→ tromagnetism) near the black hole horizons if they are cillation phases (between active species ν ν and of −→ a b charged enough. In Sec. we envisage some scenarios active into sterile species ν ν ), and consequently on a s which might result in such situation. The RNCBH may the overall neutrino dynamics (which would also include → appear, for instance, after the just-formed proto-neutron neutrino spin-ﬂip conversions ν ν , related to their l r star undergoes a phase transition creating a charge sep- handedness), have been on focus of several discussions in aration amidst the crust and the collapsing core [15], as the literature [3, 41–43]. It has become clear that grav- well as by means other possible eﬀects, which will also ity would be essential for building a complete picture of be brieﬂy discussed in Sec. . the neutrino dynamics in very dense and self-gravitating All the above motivates and leads us to the main scope matter, in particular in the very deep inside regions of su- of this work: to surmise the existence of axially symmet- pernovae. In recent analyses [3, 41], it has been pointed ric nonlinear charged black holes (at least for some in- out that the neutrino outﬂow and related supernova ex- stants of time, i.e. transiently) and study their properties pansion have been discussed in most of the speciﬁc litera- and implications, specially through the physics of neutri- ture ignoring any gravitationally-induced eﬀects. Indeed, nos. Foremost, we solve the system of equations coming neutrino oscillations in the accretion disk produced by from the minimal coupling of standard general relativity the coalescence of a binary neutron star system [2], and in with generalizations of the Maxwell’s Lagrangian, known the inner edge of the fall-back supernova ejecta, as well as as nonlinear electrodynamics (NLED), in the slow rota- around the neutrinosphere, could be strongly inﬂuenced (cid:28) tion regime (a M, a the rotational parameter of the by gravity, which then would aﬀect the supernova accre- † black hole and M its mass). tion dynamics (neutron digging in ﬁngers) [44, 45] and NLED is the approach to describe electromagnetic the ﬁnal explosion wind. Because of this, all the funda- interactions in a relativistically invariant set up. Sev- mental quantities of relevance for the explosion dynamics eral approaches were envisioned: Heisenberg; Euler and would be in principle aﬀected by the gravitational ﬁeld of Kochel; Euler; Heisenberg and Euler (added F2-term) the putatively just formed RNCBH here under analysis. [17–21]; and Weisskopf (added a logarithmic-term) [22]; Born; Born and Infeld [23] (bounded the electric ﬁeld strength by giving to the electron a ﬁnite radius), Pleban- † ski (robust framework, including plasma physics) [24], to For non experts in the ﬁeld of numerical modeling of supernovae, in what follows we resume the relevant physics and astrophysics extend Maxwell electrodynamics (linear in Lorentz in- pertinent to the concept of neutron ﬁngers taking beneﬁt from variant F) so as to deal with divergences in analysis the abstract of Ref. [44], and the discussion on this ﬂuid feature of electromagnetic (EM) phenomena, as well as to insert given in Ref. [45]. Neutron ﬁngers are instabilities in a Ledoux desired eﬀects into the theory under the classical point of stable ﬂuid driven by thermal and lepton diﬀusion, technically view. Applications of NLED have been extensively stud- quoted as doubly diﬀusive instabilities. Whenever these ﬂuid motions are present below the neutrino sphere in a core-collapse ied in the literature, extending from cosmological and supernova progenitor, they can induce convective-like ﬂuid mo- astrophysical contexts [25–32], to nonlinear optics [33], tions at those supernova layers, and may enhance the neutrino high power laser technology and plasma physics [34–37], emission by advecting neutrinos outward toward the neutrino and the ﬁeld nonlinear exponential growth due to chi- sphere, what may thus play an important role in the supernova mechanism. Neutron ﬁngers have also been suggested as being ral plasma instability during the weak parity-violating critical for producing explosions in the sophisticated spherically electron-capture (chirality imbalance) process in core symmetric supernova simulations by the Livermore group. Such collapse SNe [38, 39]. These authors stress that the orig- instability has been argued to arise in an extensive region below inal B-ﬁeld gives a positive feedback to itself, to grow the neutrino sphere of a proto-supernova where entropy and lep- ton gradients are stabilizing and destabilizing, respectively, if, as that group asserts, the rate of neutrino-mediated thermal equili- bration greatly exceeds that of neutrino-mediated lepton equili- bration. According to Bruenn and collaborators [44], application gested by QCD physics involving the appearance of Cooper of the Livermore group’s criteria to models derived from core col- pairs, “bosonization” and/or kaon condensation, color-ﬂavor lapse simulations using both their equation of state and the very locking, quark deconﬁnement, and other theoretically allowed well-known Lattimer-Swesty equation of state do show a large QCD stages which might drive phase transitions at the very inner region below the neutrinosphere unstable to neutron ﬁngers. In- core (for further details see [14] and references therein). These deed, from the convective regions below the neutrinosphere, neu- stages could appear when a equation of state of cold baryon-rich tron ﬁngers dig into the star and reach its center in about one matter is considered. All this would be as well accompanied by second. Then they propagate outward to englobe almost all the the expected neutrino cooling which would decrease even more exploding star. An interesting discussion on the relevance of this the pressure of the supernova progenitor (deleptonization pro- astrophysical ﬂuid dynamics phenomenon and its timescale (1- cess), allowing this way for gravity to make the star even further 50 ms) for the production of bursts of gravitational waves during compact. the deleptonization phase of supernovae is given in Ref. [46]. ##LTLine## ##LTLine##

text: of

text: a

text: short-lived

text: rotating

text: and

text: nonlinear

text: charged

text: black

text: exponentially,

text: being

text: this

text: last

text: the

text: actual

text: chiral

text: plasma

text: in-

text: hole

text: (RNCBH)

text: that

text: is,

text: a

text: black

text: hole

text: described

text: stability.

text: In

text: our

text: understanding,

text: this

text: ﬁeld

text: increase

text: would

text: by

text: a

text: theory

text: of

text: the

text: electromagnetism

text: more

text: general

text: than

text: suggest

text: that

text: NLED

text: might

text: be

text: at

text: action

text: inside

text: just-born

text: Maxwell’s.

text: This

text: might

text: happen

text: because

text: electromagnetic

text: pulsars.

text: For

text: further

text: details

text: on

text: magnetic

text: ﬁeld

text: ampliﬁca-

text: ﬁelds

text: could

text: easily

text: surpass

text: certain

text: scale

text: ﬁelds

text: (intrin-

text: tion,

text: see

text: Ref.

text: sically

text: associated

text: with

text: nonlinear

text: theories

text: of

text: the

text: elec-

text: Meanwhile,

text: the

text: gravitational

text: eﬀects

text: on

text: the

text: neutrino

text: os-

text: tromagnetism)

text: near

text: the

text: black

text: hole

text: horizons

text: if

text: they

text: are

text: cillation

text: phases

text: (between

text: active

text: species

text: ν

text: ν

text: and

text: of

text: a

text: b

text: charged

text: enough.

text: In

text: Sec.

text: we

text: envisage

text: some

text: scenarios

text: active

text: into

text: sterile

text: species

text: ν

text: ν

text: and

text: consequently

text: on

text: a

text: s

text: which

text: might

text: result

text: in

text: such

text: situation.

text: The

text: RNCBH

text: may

text: the

text: overall

text: neutrino

text: dynamics

text: (which

text: would

text: also

text: include

text: appear,

text: for

text: instance,

text: after

text: the

text: just-formed

text: proto-neutron

text: neutrino

text: spin-ﬂip

text: conversions

text: ν

text: ν

text: related

text: to

text: their

text: l

text: r

text: star

text: undergoes

text: a

text: phase

text: transition

text: creating

text: a

text: charge

text: sep-

text: handedness),

text: have

text: been

text: on

text: focus

text: of

text: several

text: discussions

text: in

text: aration

text: amidst

text: the

text: crust

text: and

text: the

text: collapsing

text: core

text: as

text: the

text: literature

text: It

text: has

text: become

text: clear

text: that

text: grav-

text: well

text: as

text: by

text: means

text: other

text: possible

text: eﬀects,

text: which

text: will

text: also

text: ity

text: would

text: be

text: essential

text: for

text: building

text: a

text: complete

text: picture

text: of

text: be

text: brieﬂy

text: discussed

text: in

text: Sec.

text: the

text: neutrino

text: dynamics

text: in

text: very

text: dense

text: and

text: self-gravitating

text: All

text: the

text: above

text: motivates

text: and

text: leads

text: us

text: to

text: the

text: main

text: scope

text: matter,

text: in

text: particular

text: in

text: the

text: very

text: deep

text: inside

text: regions

text: of

text: su-

text: of

text: this

text: work:

text: to

text: surmise

text: the

text: existence

text: of

text: axially

text: symmet-

text: pernovae.

text: In

text: recent

text: analyses

text: it

text: has

text: been

text: pointed

text: ric

text: nonlinear

text: charged

text: black

text: holes

text: (at

text: least

text: for

text: some

text: in-

text: out

text: that

text: the

text: neutrino

text: outﬂow

text: and

text: related

text: supernova

text: ex-

text: stants

text: of

text: time,

text: i.e.

text: transiently)

text: and

text: study

text: their

text: properties

text: pansion

text: have

text: been

text: discussed

text: in

text: most

text: of

text: the

text: speciﬁc

text: litera-

text: and

text: implications,

text: specially

text: through

text: the

text: physics

text: of

text: neutri-

text: ture

text: ignoring

text: any

text: gravitationally-induced

text: eﬀects.

text: Indeed,

text: nos.

text: Foremost,

text: we

text: solve

text: the

text: system

text: of

text: equations

text: coming

text: neutrino

text: oscillations

text: in

text: the

text: accretion

text: disk

text: produced

text: by

text: from

text: the

text: minimal

text: coupling

text: of

text: standard

text: general

text: relativity

text: the

text: coalescence

text: of

text: a

text: binary

text: neutron

text: star

text: system

text: and

text: in

text: with

text: generalizations

text: of

text: the

text: Maxwell’s

text: Lagrangian,

text: known

text: the

text: inner

text: edge

text: of

text: the

text: fall-back

text: supernova

text: ejecta,

text: as

text: well

text: as

text: as

text: nonlinear

text: electrodynamics

text: (NLED),

text: in

text: the

text: slow

text: rota-

text: around

text: the

text: neutrinosphere,

text: could

text: be

text: strongly

text: inﬂuenced

text: tion

text: regime

text: (a

text: M,

text: a

text: the

text: rotational

text: parameter

text: of

text: the

text: by

text: gravity,

text: which

text: then

text: would

text: aﬀect

text: the

text: supernova

text: accre-

text: black

text: hole

text: and

text: M

text: its

text: mass).

text: tion

text: dynamics

text: (neutron

text: digging

text: in

text: ﬁngers)

text: and

text: NLED

text: is

text: the

text: approach

text: to

text: describe

text: electromagnetic

text: the

text: ﬁnal

text: explosion

text: wind.

text: Because

text: of

text: this,

text: all

text: the

text: funda-

text: interactions

text: in

text: a

text: relativistically

text: invariant

text: set

text: up.

text: Sev-

text: mental

text: quantities

text: of

text: relevance

text: for

text: the

text: explosion

text: dynamics

text: eral

text: approaches

text: were

text: envisioned:

text: Heisenberg;

text: Euler

text: and

text: would

text: be

text: in

text: principle

text: aﬀected

text: by

text: the

text: gravitational

text: ﬁeld

text: of

text: Kochel;

text: Euler;

text: Heisenberg

text: and

text: Euler

text: (added

text: the

text: putatively

text: just

text: formed

text: RNCBH

text: here

text: under

text: analysis.

text: and

text: Weisskopf

text: (added

text: a

text: logarithmic-term)

text: Born;

text: Born

text: and

text: Infeld

text: (bounded

text: the

text: electric

text: ﬁeld

text: strength

text: by

text: giving

text: to

text: the

text: electron

text: a

text: ﬁnite

text: radius),

text: Pleban-

text: ski

text: (robust

text: framework,

text: including

text: plasma

text: physics)

text: to

text: For

text: non

text: experts

text: in

text: the

text: ﬁeld

text: of

text: numerical

text: modeling

text: of

text: supernovae,

text: in

text: what

text: follows

text: we

text: resume

text: the

text: relevant

text: physics

text: and

text: astrophysics

text: extend

text: Maxwell

text: electrodynamics

text: (linear

text: in

text: Lorentz

text: in-

text: pertinent

text: to

text: the

text: concept

text: of

text: neutron

text: ﬁngers

text: taking

text: beneﬁt

text: from

text: variant

text: F)

text: so

text: as

text: to

text: deal

text: with

text: divergences

text: in

text: analysis

text: the

text: abstract

text: of

text: Ref.

text: and

text: the

text: discussion

text: on

text: this

text: ﬂuid

text: feature

text: of

text: electromagnetic

text: (EM)

text: phenomena,

text: as

text: well

text: as

text: to

text: insert

text: given

text: in

text: Ref.

text: Neutron

text: ﬁngers

text: are

text: instabilities

text: in

text: a

text: Ledoux

text: desired

text: eﬀects

text: into

text: the

text: theory

text: under

text: the

text: classical

text: point

text: of

text: stable

text: ﬂuid

text: driven

text: by

text: thermal

text: and

text: lepton

text: diﬀusion,

text: technically

text: view.

text: Applications

text: of

text: NLED

text: have

text: been

text: extensively

text: stud-

text: quoted

text: as

text: doubly

text: diﬀusive

text: instabilities.

text: Whenever

text: these

text: ﬂuid

text: motions

text: are

text: present

text: below

text: the

text: neutrino

text: sphere

text: in

text: a

text: core-collapse

text: ied

text: in

text: the

text: literature,

text: extending

text: from

text: cosmological

text: and

text: supernova

text: progenitor,

text: they

text: can

text: induce

text: convective-like

text: ﬂuid

text: mo-

text: astrophysical

text: contexts

text: to

text: nonlinear

text: optics

text: tions

text: at

text: those

text: supernova

text: layers,

text: and

text: may

text: enhance

text: the

text: neutrino

text: high

text: power

text: laser

text: technology

text: and

text: plasma

text: physics

text: emission

text: by

text: advecting

text: neutrinos

text: outward

text: toward

text: the

text: neutrino

text: and

text: the

text: ﬁeld

text: nonlinear

text: exponential

text: growth

text: due

text: to

text: chi-

text: sphere,

text: what

text: may

text: thus

text: play

text: an

text: important

text: role

text: in

text: the

text: supernova

text: mechanism.

text: Neutron

text: ﬁngers

text: have

text: also

text: been

text: suggested

text: as

text: being

text: ral

text: plasma

text: instability

text: during

text: the

text: weak

text: parity-violating

text: critical

text: for

text: producing

text: explosions

text: in

text: the

text: sophisticated

text: spherically

text: electron-capture

text: (chirality

text: imbalance)

text: process

text: in

text: core

text: symmetric

text: supernova

text: simulations

text: by

text: the

text: Livermore

text: group.

text: Such

text: collapse

text: SNe

text: These

text: authors

text: stress

text: that

text: the

text: orig-

text: instability

text: has

text: been

text: argued

text: to

text: arise

text: in

text: an

text: extensive

text: region

text: below

text: inal

text: B-ﬁeld

text: gives

text: a

text: positive

text: feedback

text: to

text: itself,

text: to

text: grow

text: the

text: neutrino

text: sphere

text: of

text: a

text: proto-supernova

text: where

text: entropy

text: and

text: lep-

text: ton

text: gradients

text: are

text: stabilizing

text: and

text: destabilizing,

text: respectively,

text: if,

text: as

text: that

text: group

text: asserts,

text: the

text: rate

text: of

text: neutrino-mediated

text: thermal

text: equili-

text: bration

text: greatly

text: exceeds

text: that

text: of

text: neutrino-mediated

text: lepton

text: equili-

text: bration.

text: According

text: to

text: Bruenn

text: and

text: collaborators

text: application

text: gested

text: by

text: QCD

text: physics

text: involving

text: the

text: appearance

text: of

text: Cooper

text: of

text: the

text: Livermore

text: group’s

text: criteria

text: to

text: models

text: derived

text: from

text: core

text: col-

text: pairs,

text: “bosonization”

text: and/or

text: kaon

text: condensation,

text: color-ﬂavor

text: lapse

text: simulations

text: using

text: both

text: their

text: equation

text: of

text: state

text: and

text: the

text: very

text: locking,

text: quark

text: deconﬁnement,

text: and

text: other

text: theoretically

text: allowed

text: well-known

text: Lattimer-Swesty

text: equation

text: of

text: state

text: do

text: show

text: a

text: large

text: QCD

text: stages

text: which

text: might

text: drive

text: phase

text: transitions

text: at

text: the

text: very

text: inner

text: region

text: below

text: the

text: neutrinosphere

text: unstable

text: to

text: neutron

text: ﬁngers.

text: In-

text: core

text: (for

text: further

text: details

text: see

text: and

text: references

text: therein).

text: These

text: deed,

text: from

text: the

text: convective

text: regions

text: below

text: the

text: neutrinosphere,

text: neu-

text: stages

text: could

text: appear

text: when

text: a

text: equation

text: of

text: state

text: of

text: cold

text: baryon-rich

text: tron

text: ﬁngers

text: dig

text: into

text: the

text: star

text: and

text: reach

text: its

text: center

text: in

text: about

text: one

text: matter

text: is

text: considered.

text: All

text: this

text: would

text: be

text: as

text: well

text: accompanied

text: by

text: second.

text: Then

text: they

text: propagate

text: outward

text: to

text: englobe

text: almost

text: all

text: the

text: the

text: expected

text: neutrino

text: cooling

text: which

text: would

text: decrease

text: even

text: more

text: exploding

text: star.

text: An

text: interesting

text: discussion

text: on

text: the

text: relevance

text: of

text: this

text: the

text: pressure

text: of

text: the

text: supernova

text: progenitor

text: (deleptonization

text: pro-

text: astrophysical

text: ﬂuid

text: dynamics

text: phenomenon

text: and

text: its

text: timescale

text: cess),

text: allowing

text: this

text: way

text: for

text: gravity

text: to

text: make

text: the

text: star

text: even

text: further

text: ms)

text: for

text: the

text: production

text: of

text: bursts

text: of

text: gravitational

text: waves

text: during

text: compact.

text: the

text: deleptonization

text: phase

text: of

text: supernovae

text: is

text: given

text: in

text: Ref.

text: ##LTLine##

text: ##LTLine##


text: 16 a given R. This means that gravity has no eﬀect in the dius around 10 Km would be unstable for magnetic ﬁelds latter, which is the same as for ﬂat spacetime. Thence- larger than 1018 G–this value is estimated by using the forth, having available a comprehensive sample of type II virial theorem in astrophysics [77] and is related to ﬁelds supernovae and hypernovae exhibiting light polarization, of any nature (dipolar, poloidal or toroidal); stable sys- e.g. a supernova with noticeable asphericity and another tems are the ones in which their magnetic energies are with less or virtually spherical, might help discern on the smaller than the magnitude of their gravitational ener- role of rotation of the supernova progenitor and the just- gies. Thus, for a star collapsing as a whole leaving behind formed BH in the magnitude of the enhancement of the a charged magnetosphere (this might happen due to their r-process products. very diﬀerent natures, which would imply very diﬀerent On the other hand, the presence of an evolving elec- characteristic times of collapse), the latter presumably tric ﬁeld in the supernova ejecta could change the degree always present in highly magnetized systems [77], short- of polarization of light outcoming from the inner core. lived charged black holes could always be formed. It is As the strength of the electric ﬁeld depends on the total even possible that the crust may be left behind when the charge which is generating it, then comparing diﬀerent core collapses because it should interact more strongly degrees of polarization in samples of BH-forming type with the magnetosphere. In all cases, the typical sizes II supernovae and hypernovae may allow the disentan- related to charged spacetimes would be of the order of √ glement of their relative contributions to the total light the radius R of the star and the times they would be − ≈ ≈ polarization. This way, implementing a detailed analy- charged are around 10 4 s ( 1/ Gρ R/c for core − sis of supernova samples exhibiting light polarization and densities around 1015 g.cm 3). We plan to investigate the presence of electromagnetic ﬁelds (e.g. inferred from more carefully all these scenarios elsewhere. either Zeeman eﬀect or Stark eﬀect observations), one The above-mentioned mechanisms, as well as others, could have a tool for disentangling the role of rotation would motivate searches for (nonlinear) charged black and charge (nonlinearities of electromagnetism) from as- holes and their natural “probers” would be neutrinos, tronomical observations. given their bountiful production in any astrophysical con- Indeed, due to neutralization aspects, one would ex- text; see for instance Ref. [47]. As a by-product of this, in pect that charged black holes are mainly related to un- principle it would be possible to assess the nature of elec- stable scenarios, which would thus lead them to be short- tromagnetism (Maxwell’s or not), due to the special im- lived. In this regard, several physical mechanisms could print diﬀerent theories would have on certain phenomena, be conceived for their formation. We envisage some here such as neutrino oscillations, spin-ﬂip and r-processes in (for other mechanisms to generate black holes, not nec- supernova events, as we have analyzed in this work and essarily charged ones, see Ref. [76]). commented previously. One should bear in mind, though, Consider compact stars (neutron stars) that exhibit that ambiguities may still arise regarding probes of non- large magnetic ﬁelds and are good conductors (Goldreich- linear theories of the electromagnetism, since they are Julian’s model [77]). These systems are such that elec- intrinsically associated with (yet unknown) scale param- tric ﬁelds would also be present and would be of order eters as well as the charges and angular momenta of the (ΩR/c)B [77], where Ω is the angular frequency of the transient black holes, neutrinospheres, neutrino luminosi- star, and R and B are its radius and magnetic ﬁeld, re- ties, neutrino energies, neutrino masses, etc., which could spectively. From the above one immediately concludes all lead to overlaps in physical observables. Nonetheless, that the larger the magnetic ﬁeld the larger the electric even in spite of these diﬃculties, it is worthwhile investi- ﬁeld, whose origin would be related to a charge density § gating at least consequences of charged black holes, be- that for certain cases would lead to a net charge. For ≈ cause even if they are ﬂeeting they could be astrophysi- instance, a net charge of 1020 C (α 0.1 for stars with cally relevant. masses around a solar mass) would be related to a mag- In summary, we ﬁrst solved generically Einstein’s equa- netic ﬁeld of around 1019 G for (typical) neutron stars ≈ − ≈ tions for slowly rotating black holes minimally coupled to with ΩR/c 10 2 and R 106 cm. It is known that nonlinear Lagrangians of the electromagnetism depen- very large magnetic ﬁelds would lead compact stars to be dent upon its two local invariants. We used neutrinos unstable, possibly collapsing into black holes [78, 79]. A (in the WKB approximation) to probe some of the as- neutron star with a mass around 2 solar masses and ra- pects of these spacetimes, which may be invaluable tools to discern charged and uncharged black holes, as well as Maxwellian from nonlinear electrodynamics. The major § departures from the classical case concerning the mag- For example, the discontinuity of the normal component of the electric ﬁeld at the star’s surface would be related to the charge netic ﬁelds, the oﬀ-diagonal metric term, the precession − density BpΩRcos2 θ/(4πc) [77], Bp the magnetic ﬁeld at the of gyroscopes, the spin-ﬂip, the neutrino ﬂavor oscilla- pole of the star, which would result in a non-null net surface tion, etc. would just occur near the outer horizon of a charge. Besides, when one neglects macroscopic currents near nonlinear slowly rotating black hole because it would be the surface of the star and assumes it has a permittivity close to that one of the vacuum, the electric ﬁeld would be associated there that the spacetime properties would change more − · with the charge density Ω(cid:126) B(cid:126)/(2πc) [77], which in general would pronouncedly. Besides, kinematical eﬀects such as pre- also lead to a net charge. cessions (to be measured with gyroscope-like systems) ##LTLine## ##LTLine##

label: paragraph

text: a

text: given

text: R.

text: This

text: means

text: that

text: gravity

text: has

text: no

text: eﬀect

text: in

text: the

text: dius

text: around

text: Km

text: would

text: be

text: unstable

text: for

text: magnetic

text: ﬁelds

text: latter,

text: which

text: is

text: the

text: same

text: as

text: for

text: ﬂat

text: spacetime.

text: Thence-

text: larger

text: than

text: G–this

text: value

text: is

text: estimated

text: by

text: using

text: the

text: forth,

text: having

text: available

text: a

text: comprehensive

text: sample

text: of

text: type

text: II

text: virial

text: theorem

text: in

text: astrophysics

text: and

text: is

text: related

text: to

text: ﬁelds

text: supernovae

text: and

text: hypernovae

text: exhibiting

text: light

text: polarization,

text: of

text: any

text: nature

text: (dipolar,

text: poloidal

text: or

text: toroidal);

text: stable

text: sys-

text: e.g.

text: a

text: supernova

text: with

text: noticeable

text: asphericity

text: and

text: another

text: tems

text: are

text: the

text: ones

text: in

text: which

text: their

text: magnetic

text: energies

text: are

text: with

text: less

text: or

text: virtually

text: spherical,

text: might

text: help

text: discern

text: on

text: the

text: smaller

text: than

text: the

text: magnitude

text: of

text: their

text: gravitational

text: ener-

text: role

text: of

text: rotation

text: of

text: the

text: supernova

text: progenitor

text: and

text: the

text: just-

text: gies.

text: Thus,

text: for

text: a

text: star

text: collapsing

text: as

text: a

text: whole

text: leaving

text: behind

text: formed

text: BH

text: in

text: the

text: magnitude

text: of

text: the

text: enhancement

text: of

text: the

text: a

text: charged

text: magnetosphere

text: (this

text: might

text: happen

text: due

text: to

text: their

text: r-process

text: products.

text: very

text: diﬀerent

text: natures,

text: which

text: would

text: imply

text: very

text: diﬀerent

text: On

text: the

text: other

text: hand,

text: the

text: presence

text: of

text: an

text: evolving

text: elec-

text: characteristic

text: times

text: of

text: collapse),

text: the

text: latter

text: presumably

text: tric

text: ﬁeld

text: in

text: the

text: supernova

text: ejecta

text: could

text: change

text: the

text: degree

text: always

text: present

text: in

text: highly

text: magnetized

text: systems

text: short-

text: of

text: polarization

text: of

text: light

text: outcoming

text: from

text: the

text: inner

text: core.

text: lived

text: charged

text: black

text: holes

text: could

text: always

text: be

text: formed.

text: It

text: is

text: As

text: the

text: strength

text: of

text: the

text: electric

text: ﬁeld

text: depends

text: on

text: the

text: total

text: even

text: possible

text: that

text: the

text: crust

text: may

text: be

text: left

text: behind

text: when

text: the

text: charge

text: which

text: is

text: generating

text: it,

text: then

text: comparing

text: diﬀerent

text: core

text: collapses

text: because

text: it

text: should

text: interact

text: more

text: strongly

text: degrees

text: of

text: polarization

text: in

text: samples

text: of

text: BH-forming

text: type

text: with

text: the

text: magnetosphere.

text: In

text: all

text: cases,

text: the

text: typical

text: sizes

text: II

text: supernovae

text: and

text: hypernovae

text: may

text: allow

text: the

text: disentan-

text: related

text: to

text: charged

text: spacetimes

text: would

text: be

text: of

text: the

text: order

text: of

text: glement

text: of

text: their

text: relative

text: contributions

text: to

text: the

text: total

text: light

text: the

text: radius

text: R

text: of

text: the

text: star

text: and

text: the

text: times

text: they

text: would

text: be

text: polarization.

text: This

text: way,

text: implementing

text: a

text: detailed

text: analy-

text: charged

text: are

text: around

text: s

text: Gρ

text: R/c

text: for

text: core

text: sis

text: of

text: supernova

text: samples

text: exhibiting

text: light

text: polarization

text: and

text: densities

text: around

text: g.cm

text: We

text: plan

text: to

text: investigate

text: the

text: presence

text: of

text: electromagnetic

text: ﬁelds

text: (e.g.

text: inferred

text: from

text: more

text: carefully

text: all

text: these

text: scenarios

text: elsewhere.

text: either

text: Zeeman

text: eﬀect

text: or

text: Stark

text: eﬀect

text: observations),

text: one

text: The

text: above-mentioned

text: mechanisms,

text: as

text: well

text: as

text: others,

text: could

text: have

text: a

text: tool

text: for

text: disentangling

text: the

text: role

text: of

text: rotation

text: would

text: motivate

text: searches

text: for

text: (nonlinear)

text: charged

text: black

text: and

text: charge

text: (nonlinearities

text: of

text: electromagnetism)

text: from

text: as-

text: holes

text: and

text: their

text: natural

text: “probers”

text: would

text: be

text: neutrinos,

text: tronomical

text: observations.

text: given

text: their

text: bountiful

text: production

text: in

text: any

text: astrophysical

text: con-

text: Indeed,

text: due

text: to

text: neutralization

text: aspects,

text: one

text: would

text: ex-

text: text;

text: see

text: for

text: instance

text: Ref.

text: As

text: a

text: by-product

text: of

text: this,

text: in

text: pect

text: that

text: charged

text: black

text: holes

text: are

text: mainly

text: related

text: to

text: un-

text: principle

text: it

text: would

text: be

text: possible

text: to

text: assess

text: the

text: nature

text: of

text: elec-

text: stable

text: scenarios,

text: which

text: would

text: thus

text: lead

text: them

text: to

text: be

text: short-

text: tromagnetism

text: (Maxwell’s

text: or

text: not),

text: due

text: to

text: the

text: special

text: im-

text: lived.

text: In

text: this

text: regard,

text: several

text: physical

text: mechanisms

text: could

text: print

text: diﬀerent

text: theories

text: would

text: have

text: on

text: certain

text: phenomena,

text: be

text: conceived

text: for

text: their

text: formation.

text: We

text: envisage

text: some

text: here

text: such

text: as

text: neutrino

text: oscillations,

text: spin-ﬂip

text: and

text: r-processes

text: in

text: (for

text: other

text: mechanisms

text: to

text: generate

text: black

text: holes,

text: not

text: nec-

text: supernova

text: events,

text: as

text: we

text: have

text: analyzed

text: in

text: this

text: work

text: and

text: essarily

text: charged

text: ones,

text: see

text: Ref.

text: commented

text: previously.

text: One

text: should

text: bear

text: in

text: mind,

text: though,

text: Consider

text: compact

text: stars

text: (neutron

text: stars)

text: that

text: exhibit

text: that

text: ambiguities

text: may

text: still

text: arise

text: regarding

text: probes

text: of

text: non-

text: large

text: magnetic

text: ﬁelds

text: and

text: are

text: good

text: conductors

text: (Goldreich-

text: linear

text: theories

text: of

text: the

text: electromagnetism,

text: since

text: they

text: are

text: Julian’s

text: model

text: These

text: systems

text: are

text: such

text: that

text: elec-

text: intrinsically

text: associated

text: with

text: (yet

text: unknown)

text: scale

text: param-

text: tric

text: ﬁelds

text: would

text: also

text: be

text: present

text: and

text: would

text: be

text: of

text: order

text: eters

text: as

text: well

text: as

text: the

text: charges

text: and

text: angular

text: momenta

text: of

text: the

text: (ΩR/c)B

text: where

text: Ω

text: is

text: the

text: angular

text: frequency

text: of

text: the

text: transient

text: black

text: holes,

text: neutrinospheres,

text: neutrino

text: luminosi-

text: star,

text: and

text: R

text: and

text: B

text: are

text: its

text: radius

text: and

text: magnetic

text: ﬁeld,

text: re-

text: ties,

text: neutrino

text: energies,

text: neutrino

text: masses,

text: etc.,

text: which

text: could

text: spectively.

text: From

text: the

text: above

text: one

text: immediately

text: concludes

text: all

text: lead

text: to

text: overlaps

text: in

text: physical

text: observables.

text: Nonetheless,

text: that

text: the

text: larger

text: the

text: magnetic

text: ﬁeld

text: the

text: larger

text: the

text: electric

text: even

text: in

text: spite

text: of

text: these

text: diﬃculties,

text: it

text: is

text: worthwhile

text: investi-

text: ﬁeld,

text: whose

text: origin

text: would

text: be

text: related

text: to

text: a

text: charge

text: density

text: gating

text: at

text: least

text: consequences

text: of

text: charged

text: black

text: holes,

text: be-

text: that

text: for

text: certain

text: cases

text: would

text: lead

text: to

text: a

text: net

text: charge.

text: For

text: cause

text: even

text: if

text: they

text: are

text: ﬂeeting

text: they

text: could

text: be

text: astrophysi-

text: instance,

text: a

text: net

text: charge

text: of

text: C

text: (α

text: for

text: stars

text: with

text: cally

text: relevant.

text: masses

text: around

text: a

text: solar

text: mass)

text: would

text: be

text: related

text: to

text: a

text: mag-

text: In

text: summary,

text: we

text: ﬁrst

text: solved

text: generically

text: Einstein’s

text: equa-

text: netic

text: ﬁeld

text: of

text: around

text: G

text: for

text: (typical)

text: neutron

text: stars

text: tions

text: for

text: slowly

text: rotating

text: black

text: holes

text: minimally

text: coupled

text: to

text: with

text: ΩR/c

text: and

text: R

text: cm.

text: It

text: is

text: known

text: that

text: nonlinear

text: Lagrangians

text: of

text: the

text: electromagnetism

text: depen-

text: very

text: large

text: magnetic

text: ﬁelds

text: would

text: lead

text: compact

text: stars

text: to

text: be

text: dent

text: upon

text: its

text: two

text: local

text: invariants.

text: We

text: used

text: neutrinos

text: unstable,

text: possibly

text: collapsing

text: into

text: black

text: holes

text: A

text: (in

text: the

text: WKB

text: approximation)

text: to

text: probe

text: some

text: of

text: the

text: as-

text: neutron

text: star

text: with

text: a

text: mass

text: around

text: solar

text: masses

text: and

text: ra-

text: pects

text: of

text: these

text: spacetimes,

text: which

text: may

text: be

text: invaluable

text: tools

text: to

text: discern

text: charged

text: and

text: uncharged

text: black

text: holes,

text: as

text: well

text: as

text: Maxwellian

text: from

text: nonlinear

text: electrodynamics.

text: The

text: major

text: departures

text: from

text: the

text: classical

text: case

text: concerning

text: the

text: mag-

text: For

text: example,

text: the

text: discontinuity

text: of

text: the

text: normal

text: component

text: of

text: the

text: electric

text: ﬁeld

text: at

text: the

text: star’s

text: surface

text: would

text: be

text: related

text: to

text: the

text: charge

text: netic

text: ﬁelds,

text: the

text: oﬀ-diagonal

text: metric

text: term,

text: the

text: precession

text: density

text: Bp

text: the

text: magnetic

text: ﬁeld

text: at

text: the

text: of

text: gyroscopes,

text: the

text: spin-ﬂip,

text: the

text: neutrino

text: ﬂavor

text: oscilla-

text: pole

text: of

text: the

text: star,

text: which

text: would

text: result

text: in

text: a

text: non-null

text: net

text: surface

text: tion,

text: etc.

text: would

text: just

text: occur

text: near

text: the

text: outer

text: horizon

text: of

text: a

text: charge.

text: Besides,

text: when

text: one

text: neglects

text: macroscopic

text: currents

text: near

text: nonlinear

text: slowly

text: rotating

text: black

text: hole

text: because

text: it

text: would

text: be

text: the

text: surface

text: of

text: the

text: star

text: and

text: assumes

text: it

text: has

text: a

text: permittivity

text: close

text: to

text: that

text: one

text: of

text: the

text: vacuum,

text: the

text: electric

text: ﬁeld

text: would

text: be

text: associated

text: there

text: that

text: the

text: spacetime

text: properties

text: would

text: change

text: more

text: with

text: the

text: charge

text: density

text: which

text: in

text: general

text: would

text: pronouncedly.

text: Besides,

text: kinematical

text: eﬀects

text: such

text: as

text: pre-

text: also

text: lead

text: to

text: a

text: net

text: charge.

text: cessions

text: (to

text: be

text: measured

text: with

text: gyroscope-like

text: systems)

text: ##LTLine##

text: ##LTLine##


text: 6 − as it can be shown by choosing L(F, G) = F/4 and solving the equations for the metric g , E and − 00 r0 E˜r2 + aA(r)˜l ˜lg (r) E˜aA(r) Eqs. (19) and (21). t˙ = , ϕ˙ = 00 , (24) g (r)r2 g (r)r2 Since in general the problem set out above can just be 00 00 solved numerically, it turns out that the r variable is not where we neglected terms of second order in a/r and convenient for this end. Numerically, it is much more . deﬁned that for any quantity C, C˜ = C/m. Another ﬁrst suitable to use the dimensionless variable u, deﬁned as . integral that comes out(cid:34)of our prescription i(cid:35)s u = M/r. Concerning the integration of the aforesaid equations, u should run from 0 to u , where the latter h is given as the smallest solution to g00(uh) = 0. The − ˜l2 − 2E˜˜laA(r) quantity uh is called the outermost horizon of the black r˙2 = E˜2 g00(r) r2 g (r)r2 + (cid:15) , (25) hole and determining it is important because it deﬁnes 00 the region of physical interest (u < u ), which has an h where (cid:15) = 0, 1, according to which the geodesic is light- impact on all position-dependent observables. like or time-like, respectively. The above equation is ob- III. GEODESICS IN SLOWLY ROTATING tained by means of the line element given by Eq. (4). NONLINEAR SPACETIMES Just for the sake of completeness, the last ﬁrst integral of our analysis is θ˙ = 0. From Eq. (25), one can even Forasmuch as we are interested in describing neutri- deﬁne an eﬀective potential by means of V˜ 2 = E˜2, for nos in spacetimes related to axially symmetric nonlinear r˙ = 0, which then reads [49(cid:118)(cid:117), 55] (cid:34) (cid:35) black holes, the general study of geodesics is necessary, (cid:117) (cid:116) given that these particles have no charge and hence are not sensitive to forces of electromagnetic origin. This is ˜laA(r) ± ˜l2 also so since the Dirac equation in the limit of the WKB V˜± = g (r) + (cid:15) . (26) r2 00 r2 approximation (the one we shall be interested in here) assures that the phase part of neutrino spinors satisfy a Hamilton-Jacobi-like equation [52] and by assuming that One could just work with V˜ , since the “symmetry − − + their amplitudes vary slowly, they do not play a role for rule” V˜−(˜l) = V˜ ( ˜l) holds [55]. All features charac- + convenient spacetime distances, revealing therefore the terizing the motion of neutral test particles can be ob- test-particle aspects of the neutrinos (quite similarly to tained by means of the scrutiny of the above equation. the notion of rays in optics). We underline from the pre- We shall not perform such an analysis here, for we are vious sentences that we are overlooking the interaction only interested in neutrino spin-ﬂip transitions and ﬂavor of the B-ﬁeld with the neutrino anomalous magnetic mo- oscillations. ment and spin. Upon the aforesaid premises, there are In what follows, we shall discuss neutrino oscillations −→ several ways of describing neutrino trajectories. An ele- (ν ν ) in connection to the oscillation length and a b gant approach would be solving the associated Hamilton- the transition probability. Analysis regarding spin-ﬂip Jacobi equation [51] for the spacetime given by Eq. (4). will be done in Sec. . Nevertheless, we will follow the Lagrangian approach. IV. NEUTRINO FLAVOR OSCILLATION Given that we are working up to ﬁrst order in “a/r”, due to the frame dragging eﬀect, test-particles only re- As stated previously, neutrino ﬂavor oscillations would main conﬁned in a plane if it is the equatorial one [53], take place due to the fact that neutrino ﬂavor eigenstates | (cid:105) and thus for simplicity we limit our analysis to θ = π/2. ν are linear combinations of neutrino mass eigenstates | α(cid:105) For this case, the proper Lagrangian for test particles ν as (see e.g. [56] and references therein) j (t.p.) is [54] (cid:20) (cid:21) | (cid:105) − | (cid:105) ν = U exp[ iΦ ] ν , (27) α αj j j . 1 − r˙2 − − L(t.p.) = g t˙2 r2ϕ˙2 2aA(r)t˙ϕ˙ , (23) 2 00 g where repeated indexes are summed over. In the above 00 equation, the α index stands for the neutrino ﬂavor eigen- states, while the j one stands for the masses eigenstates. . where x˙µ = dxµ/dλ, with λ an aﬃne parameter along The matrix Uαj is a unitary matrix that gives the mix- the curve followed by the test particle. From Eq. (4), ing - level-crossing - between the ﬂavor eigenstates and the coordinates t and ϕ are cyclic on. es for the a.bo−ve the mass eigenstates. Besides, Φj is the phase associated Lagrangian. Hence, t.he quantities pt = E and pϕ = l, with the jth mass eigenst(cid:90)ate. For curved spacetimes, Φj with p = g pµ, pµ = mx˙µ, m the rest mass of the test reads [56] µ µν particle of interest, are constants along the geodesics. From our previous deﬁnitions and Eq. (23), we thus Φ = P dxµ, (28) have j (j)µ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: as

text: it

text: can

text: be

text: shown

text: by

text: choosing

text: L(F,

text: G)

text: and

text: solving

text: the

text: equations

text: for

text: the

text: metric

text: g

text: E

text: and

label: equation

text: aA(r)˜l

text: ˜lg

text: (r)

text: E˜aA(r)

text: Eqs.

text: and

text: t˙

text: ϕ˙

text: g

text: g

text: Since

text: in

text: general

text: the

text: problem

text: set

text: out

text: above

text: can

text: just

text: be

text: solved

text: numerically,

text: it

text: turns

text: out

text: that

text: the

text: r

text: variable

text: is

text: not

text: where

text: we

text: neglected

text: terms

text: of

text: second

text: order

text: in

text: a/r

text: and

text: convenient

text: for

text: this

text: end.

text: Numerically,

text: it

text: is

text: much

text: more

text: deﬁned

text: that

text: for

text: any

text: quantity

text: C,

text: C˜

text: C/m.

text: Another

text: ﬁrst

text: suitable

text: to

text: use

text: the

text: dimensionless

text: variable

text: u,

text: deﬁned

text: as

text: integral

text: that

text: comes

text: our

text: prescription

text: u

text: M/r.

text: Concerning

text: the

text: integration

text: of

text: the

text: aforesaid

text: equations,

text: u

text: should

text: run

text: from

text: to

text: u

text: where

text: the

text: latter

text: h

text: is

text: given

text: as

text: the

text: smallest

text: solution

text: to

text: The

text: quantity

text: uh

text: is

text: called

text: the

text: outermost

text: horizon

text: of

text: the

text: black

text: g

text: hole

text: and

text: determining

text: it

text: is

text: important

text: because

text: it

text: deﬁnes

text: the

text: region

text: of

text: physical

text: interest

text: (u

text: u

text: which

text: has

text: an

text: h

text: where

text: according

text: to

text: which

text: the

text: geodesic

text: is

text: light-

text: impact

text: on

text: all

text: position-dependent

text: observables.

text: like

text: or

text: time-like,

text: respectively.

text: The

text: above

text: equation

text: is

text: ob-

text: III.

text: GEODESICS

text: IN

text: SLOWLY

text: ROTATING

text: tained

text: by

text: means

text: of

text: the

text: line

text: element

text: given

text: by

text: Eq.

text: NONLINEAR

text: SPACETIMES

text: Just

text: for

text: the

text: sake

text: of

text: completeness,

text: the

text: last

text: ﬁrst

text: integral

text: of

text: our

text: analysis

text: is

text: θ˙

text: From

text: Eq.

text: one

text: can

text: even

text: Forasmuch

text: as

text: we

text: are

text: interested

text: in

text: describing

text: neutri-

text: deﬁne

text: an

text: eﬀective

text: potential

text: by

text: means

text: of

text: V˜

text: for

text: nos

text: in

text: spacetimes

text: related

text: to

text: axially

text: symmetric

text: nonlinear

text: r˙

text: which

text: then

text: reads

text: black

text: holes,

text: the

text: general

text: study

text: of

text: geodesics

text: is

text: necessary,

text: given

text: that

text: these

text: particles

text: have

text: no

text: charge

text: and

text: hence

text: are

text: not

text: sensitive

text: to

text: forces

text: of

text: electromagnetic

text: origin.

text: This

text: is

text: ˜laA(r)

text: also

text: so

text: since

text: the

text: Dirac

text: equation

text: in

text: the

text: limit

text: of

text: the

text: WKB

text: V˜±

text: g

text: (r)

text: approximation

text: (the

text: one

text: we

text: shall

text: be

text: interested

text: in

text: here)

text: assures

text: that

text: the

text: phase

text: part

text: of

text: neutrino

text: spinors

text: satisfy

text: a

text: Hamilton-Jacobi-like

text: equation

text: and

text: by

text: assuming

text: that

text: One

text: could

text: just

text: work

text: with

text: V˜

text: since

text: the

text: “symmetry

text: their

text: amplitudes

text: vary

text: slowly,

text: they

text: do

text: not

text: play

text: a

text: role

text: for

text: rule”

text: V˜−(˜l)

text: V˜

text: ˜l)

text: holds

text: All

text: features

text: charac-

text: convenient

text: spacetime

text: distances,

text: revealing

text: therefore

text: the

text: terizing

text: the

text: motion

text: of

text: neutral

text: test

text: particles

text: can

text: be

text: ob-

text: test-particle

text: aspects

text: of

text: the

text: neutrinos

text: (quite

text: similarly

text: to

text: tained

text: by

text: means

text: of

text: the

text: scrutiny

text: of

text: the

text: above

text: equation.

text: the

text: notion

text: of

text: rays

text: in

text: optics).

text: We

text: underline

text: from

text: the

text: pre-

text: We

text: shall

text: not

text: perform

text: such

text: an

text: analysis

text: here,

text: for

text: we

text: are

text: vious

text: sentences

text: that

text: we

text: are

text: overlooking

text: the

text: interaction

text: only

text: interested

text: in

text: neutrino

text: spin-ﬂip

text: transitions

text: and

text: ﬂavor

text: of

text: the

text: B-ﬁeld

text: with

text: the

text: neutrino

text: anomalous

text: magnetic

text: mo-

text: oscillations.

text: ment

text: and

text: spin.

text: Upon

text: the

text: aforesaid

text: premises,

text: there

text: are

text: In

text: what

text: follows,

text: we

text: shall

text: discuss

text: neutrino

text: oscillations

text: several

text: ways

text: of

text: describing

text: neutrino

text: trajectories.

text: An

text: ele-

text: (ν

text: ν

text: in

text: connection

text: to

text: the

text: oscillation

text: length

text: and

text: a

text: b

text: gant

text: approach

text: would

text: be

text: solving

text: the

text: associated

text: Hamilton-

text: the

text: transition

text: probability.

text: Analysis

text: regarding

text: spin-ﬂip

text: Jacobi

text: equation

text: for

text: the

text: spacetime

text: given

text: by

text: Eq.

text: will

text: be

text: done

text: in

text: Sec.

text: Nevertheless,

text: we

text: will

text: follow

text: the

text: Lagrangian

text: approach.

text: IV.

text: NEUTRINO

text: FLAVOR

text: OSCILLATION

text: Given

text: that

text: we

text: are

text: working

text: up

text: to

text: ﬁrst

text: order

text: in

text: “a/r”,

text: due

text: to

text: the

text: frame

text: dragging

text: eﬀect,

text: test-particles

text: only

text: re-

text: As

text: stated

text: previously,

text: neutrino

text: ﬂavor

text: oscillations

text: would

text: main

text: conﬁned

text: in

text: a

text: plane

text: if

text: it

text: is

text: the

text: equatorial

text: one

text: take

text: place

text: due

text: to

text: the

text: fact

text: that

text: neutrino

text: ﬂavor

text: eigenstates

text: and

text: thus

text: for

text: simplicity

text: we

text: limit

text: our

text: analysis

text: to

text: θ

text: ν

text: are

text: linear

text: combinations

text: of

text: neutrino

text: mass

text: eigenstates

text: For

text: this

text: case,

text: the

text: proper

text: Lagrangian

text: for

text: test

text: particles

text: ν

text: as

text: (see

text: e.g.

text: and

text: references

text: therein)

text: j

text: (t.p.)

text: is

text: ν

text: U

text: exp[

text: iΦ

text: ν

text: α

text: αj

text: j

text: j

text: L(t.p.)

text: g

text: g

text: where

text: repeated

text: indexes

text: are

text: summed

text: over.

text: In

text: the

text: above

text: equation,

text: the

text: α

text: index

text: stands

text: for

text: the

text: neutrino

text: ﬂavor

text: eigen-

text: states,

text: while

text: the

text: j

text: one

text: stands

text: for

text: the

text: masses

text: eigenstates.

text: where

text: x˙µ

text: dxµ/dλ,

text: with

text: λ

text: an

text: aﬃne

text: parameter

text: along

text: The

text: matrix

text: Uαj

text: is

text: a

text: unitary

text: matrix

text: that

text: gives

text: the

text: mix-

text: the

text: curve

text: followed

text: by

text: the

text: test

text: particle.

text: From

text: Eq.

text: ing

text: level-crossing

text: between

text: the

text: ﬂavor

text: eigenstates

text: and

text: the

text: coordinates

text: t

text: and

text: ϕ

text: are

text: cyclic

text: on.

text: es

text: for

text: the

text: a.bo−ve

text: the

text: mass

text: eigenstates.

text: Besides,

text: Φj

text: is

text: the

text: phase

text: associated

text: Lagrangian.

text: Hence,

text: t.he

text: quantities

text: pt

text: E

text: and

text: pϕ

text: l,

text: with

text: the

text: jth

text: mass

text: For

text: curved

text: spacetimes,

text: Φj

text: with

text: p

text: g

text: pµ,

text: pµ

text: mx˙µ,

text: m

text: the

text: rest

text: mass

text: of

text: the

text: test

text: reads

text: µ

text: µν

text: particle

text: of

text: interest,

text: are

text: constants

text: along

text: the

text: geodesics.

text: From

text: our

text: previous

text: deﬁnitions

text: and

text: Eq.

text: we

text: thus

text: Φ

text: P

text: dxµ,

text: have

text: j

text: (j)µ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 Another important quantity that arises in the descrip- where P is just to indicate the four-momentum of tion of neutrino oscillations is the oscillation length [58, (j)µ the mass eigenstate j. We shall assume that neutrinos 60]. Basically it estimates the length over which a given just have two spin ﬂavors. It is well-known [57] in this neutrino has to travel for Φ to change by 2π. Therefore, jk case that one can introduce a mixing angle, Θ, such that for talking about oscillation lengths, the proper relativis- the transition probability from one ﬂavor eigenstate α to tic covariant distance is of importance [43]. For the case another β reads (cid:18) (cid:19) of the spacetimes described by Eq. (4), from Eqs. (24) and (25) and assuming that the particles involved have (cid:29) the same energy E and are such that E m , it follows j,k P(ν → ν ) = sin2(2Θ) sin2 Φjk , (29) that α β 2 where Φjk =. Φj − Φk. Whenever one is interested Losc =. dΦ dl/p(r2π) (cid:39) √g (2mπ2E− m2), (31) in neutrino propagation in spacetimes given by Eq. (4), jk 00 j k after Eqs. (24) and (25) are taken into account, for the (cid:54) where we assumed that dl is the inﬁnitesimal proper case r˙ = 0, E(cid:90)q. (28) can be cast into the form pr distance, given by [(cid:18)51] (cid:19) (cid:90) m (cid:15) Φj = (cid:114)dr r˙j = (cid:104) (cid:105) dlp2r = −gij + g0gig0j dxidxj, (32) 00 (cid:15)dr m2 . (30) j − − E2 g (r) l2 2ElaA(r) + m2(cid:15) with i, j = 1, 2, 3. If one wants to restore the conven- 00 r2 g00(r)r2 j tional units, the right-hand side of the equality in Eq.(31) (cid:126) must be multiplied by /c3. Note that Eq. (30) is exact and becomes zero for Hence, from Eq. (31) we learn that the oscillation null geodesics. This is e∝asily understood by the fact length decreases whenever g00 increases. This is exactly that p dxµ = g pβdxµ ds2, which is zero for null the case for nonlinear charged black holes, when com- µ µβ paths. Hence, when it is stated in the literature that pared to a Schwarzschild black hole. This means that null paths are taken into consideration, approximations when the black hole is charged, neutrinos will tend to are done such that in parts of the Eq. (30) properties oscillate more than they would when it is not charged, of null geodesic are utilized. (In the standard treatment for each spacetime point (location). one assumes that p = g pν is deﬁned along a time-like V. NEUTRINO SPIN PRECESSION µ µν geodesic, while the tangent four-vector to the trajectory dxµ/dλ is taken to be null-like, resulting in a factor of 2 In this section we summarize the main points about when compared to the case both p and x˙µ are deﬁned neutrino ﬂavor spin precession, also named neutrino ﬂa- µ along time-like geodesics. See [58] for further details.) In vor spin-ﬂip, or neutrino-antineutrino oscillations [61], the nonlinear case, it is momentous to bear in mind that and study them in the framework of the metric given photons do not follow null geodesics in their background by Eq. (4). For point-like particles, the equations gov- spacetimes, but in the so-called eﬀective geometries (see erning the spin Sµ coupling of test particles with the e.g. [59] and references therein). Therefore, the distinc- gravitational ﬁeld are [1] tion between massive particles and photons in our case is paramount. DSµ Duµ From the second expression in Eq. (24) we see that = 0, = 0, (33) dλ dλ in general it is impossible to have ϕ˙ = 0. Hence, pure radial geodesics do not exist in axially symmetric space- times. The origin for so is the dragging of inertial frames, where D/dλ stands for the absolute derivative with also known as Lense-Thirring eﬀect [49]. Nevertheless, λ an aﬃne parameter [51]. From the deﬁnition of the ∼ O approximating ϕ˙ (a) is always possible if one as- absolute derivative, one sees that the spin does change ∼ O sumes that ˜l (a). For these particular (nearly ra- whenever spin connections are not null, as contrary to the case of an intrinsically ﬂat Minkowski spacetime. dial) geodesics, the eﬀects introduced by the nonlineari- A proper analysis about the spin evolution of a test ties of the Lagrangians are completely washed away, since (cid:29) ≤ particle by a local observer is done with the use of (or- E m, and 0 < g 1 outside the horizon. Hence, 00 thonormal) tetrads (ea), i.e., [51] although the neutrino oscillation expression is that from µ special relativity in this case, the general relativistic ef- fect of frame dragging still persists on their trajectories. g = η eaeb, eaeν = δν , eaeµ = δa, (34) A. Neutrino oscillation length µν ab µ ν µ a µ µ b b ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Another

text: important

text: quantity

text: that

text: arises

text: in

text: the

text: descrip-

text: where

text: P

text: is

text: just

text: to

text: indicate

text: the

text: four-momentum

text: of

text: tion

text: of

text: neutrino

text: oscillations

text: is

text: the

text: oscillation

text: length

text: (j)µ

text: the

text: mass

text: eigenstate

text: j.

text: We

text: shall

text: assume

text: that

text: neutrinos

text: Basically

text: it

text: estimates

text: the

text: length

text: over

text: which

text: a

text: given

text: just

text: have

text: two

text: spin

text: ﬂavors.

text: It

text: is

text: well-known

text: in

text: this

text: neutrino

text: has

text: to

text: travel

text: for

text: Φ

text: to

text: change

text: by

text: Therefore,

text: jk

text: case

text: that

text: one

text: can

text: introduce

text: a

text: mixing

text: angle,

text: Θ,

text: such

text: that

text: for

text: talking

text: about

text: oscillation

text: lengths,

text: the

text: proper

text: relativis-

text: the

text: transition

text: probability

text: from

text: one

text: ﬂavor

text: eigenstate

text: α

text: to

text: tic

text: covariant

text: distance

text: is

text: of

text: importance

text: For

text: the

text: case

text: another

text: β

text: reads

text: of

text: the

text: spacetimes

text: described

text: by

text: Eq.

text: from

text: Eqs.

text: and

text: and

text: assuming

text: that

text: the

text: particles

text: involved

text: have

text: the

text: same

text: energy

text: E

text: and

text: are

text: such

text: that

text: E

text: m

text: it

text: follows

text: j,k

text: P(ν

text: ν

text: Φjk

text: that

text: α

text: β

text: where

text: Φjk

text: Φj

text: Φk.

text: Whenever

text: one

text: is

text: interested

text: Losc

text: dΦ

text: √g

text: in

text: neutrino

text: propagation

text: in

text: spacetimes

text: given

text: by

text: Eq.

text: jk

text: j

text: k

text: after

text: Eqs.

text: and

text: are

text: taken

text: into

text: account,

text: for

text: the

text: where

text: we

text: assumed

text: that

text: dl

text: is

text: the

text: inﬁnitesimal

text: proper

text: case

text: r˙

text: can

text: be

text: cast

text: into

text: the

text: form

text: pr

text: distance,

text: given

text: by

text: m

text: Φj

text: r˙j

text: −gij

text: dxidxj,

text: j

text: g

text: (r)

text: with

text: i,

text: j

text: If

text: one

text: wants

text: to

text: restore

text: the

text: conven-

text: j

text: tional

text: units,

text: the

text: right-hand

text: side

text: of

text: the

text: equality

text: in

text: must

text: be

text: multiplied

text: by

text: Note

text: that

text: Eq.

text: is

text: exact

text: and

text: becomes

text: zero

text: for

text: Hence,

text: from

text: Eq.

text: we

text: learn

text: that

text: the

text: oscillation

text: geodesics.

text: This

text: is

text: e∝asily

text: understood

text: by

text: the

text: fact

text: length

text: decreases

text: whenever

text: increases.

text: This

text: is

text: exactly

text: that

text: p

text: dxµ

text: g

text: pβdxµ

text: which

text: is

text: zero

text: for

text: the

text: case

text: for

text: nonlinear

text: charged

text: black

text: holes,

text: when

text: com-

text: µ

text: µβ

text: paths.

text: Hence,

text: when

text: it

text: is

text: stated

text: in

text: the

text: literature

text: that

text: pared

text: to

text: a

text: Schwarzschild

text: black

text: hole.

text: This

text: means

text: that

text: paths

text: are

text: taken

text: into

text: consideration,

text: approximations

text: when

text: the

text: black

text: hole

text: is

text: charged,

text: neutrinos

text: will

text: tend

text: to

text: are

text: done

text: such

text: that

text: in

text: parts

text: of

text: the

text: Eq.

text: properties

text: oscillate

text: more

text: than

text: they

text: would

text: when

text: it

text: is

text: not

text: charged,

text: of

text: geodesic

text: are

text: utilized.

text: (In

text: the

text: standard

text: treatment

text: for

text: each

text: spacetime

text: point

text: (location).

text: one

text: assumes

text: that

text: p

text: g

text: pν

text: is

text: deﬁned

text: along

text: a

text: time-like

text: V.

text: NEUTRINO

text: SPIN

text: PRECESSION

text: µ

text: µν

text: geodesic,

text: while

text: the

text: tangent

text: four-vector

text: to

text: the

text: trajectory

text: dxµ/dλ

text: is

text: taken

text: to

text: be

text: null-like,

text: resulting

text: in

text: a

text: factor

text: of

text: In

text: this

text: section

text: we

text: summarize

text: the

text: main

text: points

text: about

text: when

text: compared

text: to

text: the

text: case

text: both

text: p

text: and

text: x˙µ

text: are

text: deﬁned

text: neutrino

text: ﬂavor

text: spin

text: precession,

text: also

text: named

text: neutrino

text: ﬂa-

text: µ

text: along

text: time-like

text: geodesics.

text: See

text: for

text: further

text: details.)

text: In

text: vor

text: spin-ﬂip,

text: or

text: neutrino-antineutrino

text: oscillations

text: the

text: nonlinear

text: case,

text: it

text: is

text: momentous

text: to

text: bear

text: in

text: mind

text: that

text: and

text: study

text: them

text: in

text: the

text: framework

text: of

text: the

text: metric

text: given

text: photons

text: do

text: not

text: follow

text: geodesics

text: in

text: their

text: background

text: by

text: Eq.

text: For

text: point-like

text: particles,

text: the

text: equations

text: gov-

text: spacetimes,

text: but

text: in

text: the

text: so-called

text: eﬀective

text: geometries

text: (see

text: erning

text: the

text: spin

text: Sµ

text: coupling

text: of

text: test

text: particles

text: with

text: the

text: e.g.

text: and

text: references

text: therein).

text: Therefore,

text: the

text: distinc-

text: gravitational

text: ﬁeld

text: are

text: tion

text: between

text: massive

text: particles

text: and

text: photons

text: in

text: our

text: case

text: is

text: paramount.

text: DSµ

text: Duµ

text: From

text: the

text: second

text: expression

text: in

text: Eq.

text: we

text: see

text: that

text: dλ

text: dλ

text: in

text: general

text: it

text: is

text: impossible

text: to

text: have

text: ϕ˙

text: Hence,

text: pure

text: radial

text: geodesics

text: do

text: not

text: exist

text: in

text: axially

text: symmetric

text: space-

text: times.

text: The

text: origin

text: for

text: so

text: is

text: the

text: dragging

text: of

text: inertial

text: frames,

text: where

text: D/dλ

text: stands

text: for

text: the

text: absolute

text: derivative

text: with

text: also

text: known

text: as

text: Lense-Thirring

text: eﬀect

text: Nevertheless,

text: λ

text: an

text: aﬃne

text: parameter

text: From

text: the

text: deﬁnition

text: of

text: the

text: O

text: approximating

text: ϕ˙

text: (a)

text: is

text: always

text: possible

text: if

text: one

text: as-

text: absolute

text: derivative,

text: one

text: sees

text: that

text: the

text: spin

text: does

text: change

text: O

text: sumes

text: that

text: ˜l

text: (a).

text: For

text: these

text: particular

text: (nearly

text: ra-

text: whenever

text: spin

text: connections

text: are

text: not

text: null,

text: as

text: contrary

text: to

text: the

text: case

text: of

text: an

text: intrinsically

text: ﬂat

text: Minkowski

text: spacetime.

text: dial)

text: geodesics,

text: the

text: eﬀects

text: introduced

text: by

text: the

text: nonlineari-

text: A

text: proper

text: analysis

text: about

text: the

text: spin

text: evolution

text: of

text: a

text: test

text: ties

text: of

text: the

text: Lagrangians

text: are

text: completely

text: washed

text: away,

text: since

text: particle

text: by

text: a

text: local

text: observer

text: is

text: done

text: with

text: the

text: use

text: of

text: (or-

text: E

text: m,

text: and

text: g

text: outside

text: the

text: horizon.

text: Hence,

text: thonormal)

text: tetrads

text: (ea),

text: i.e.,

text: although

text: the

text: neutrino

text: oscillation

text: expression

text: is

text: that

text: from

text: µ

text: special

text: relativity

text: in

text: this

text: case,

text: the

text: general

text: relativistic

text: ef-

text: fect

text: of

text: frame

text: dragging

text: still

text: persists

text: on

text: their

text: trajectories.

text: g

text: η

text: eaeb,

text: eaeν

text: δν

text: eaeµ

text: δa,

text: A.

text: Neutrino

text: oscillation

text: length

text: µν

text: ab

text: µ

text: ν

text: µ

text: a

text: µ

text: µ

text: b

text: b

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: A Third-Order Moving Mesh Cell-Centered Scheme for One-Dimensional Elastic-Plastic Flows ∗ † ‡ § Jun-Bo Cheng Weizhang Huang Song Jiang Baolin Tian A third-order moving mesh cell-centered scheme without the remapping of physical variables is developed for the numerical solution of one-dimensional elastic-plastic ﬂows with the Mie-Gru¨neisen equation of state, the Wilkins constitutive model, and the von Mises yielding criterion. The scheme combines the Lagrangian method with the MMPDE moving mesh method and adaptively moves the mesh to better resolve shock and other types of waves while preventing the mesh from crossing and tangling. It can be viewed as a direct arbitrarily Lagrangian-Eulerian method but can also be degenerated to a purely Lagrangian scheme. It treats the relative velocity of the ﬂuid with respect to the mesh as constant in time between time steps, which allows high-order approximation of free boundaries. A time dependent scaling is used in the monitor function to avoid possible sudden movement of the mesh points due to the creation or diminishing of shock and rarefaction waves or the steepening of those waves. A two-rarefaction Riemann solver with elastic waves is employed to compute the Godunov values of the density, pressure, velocity, and deviatoric stress at cell interfaces. Numerical results are presented for three examples. The third-order convergence of the scheme and its ability to concentrate mesh points around shock and elastic rarefaction waves are demonstrated. The obtained numerical results are in good agreement with those in literature. The new scheme is also shown to be more accurate in resolving shock and rarefaction waves than an existing third-order cell-centered Lagrangian scheme. Key Words. Elastic-plastic ﬂows; Cell-centered Lagrangian scheme; Moving mesh method; High order scheme; Hypoelastic constitutive model Abbreviated title. A third-order moving mesh scheme for elastic-plastic ﬂows ∗ Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China (cheng junbo@iapcm.ac.cn) † Department of Mathematics, the University of Kansas, Lawrence, KS 66045, U.S.A. (whuang@ku.edu) ‡ Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China (jiang@iapcm.ac.cn) § Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, China (tian baolin@iapcm.ac.cn) 1 ##LTLine## ##LTLine## ##LTLine##

text: A

label: title

text: Third-Order

text: Moving

text: Mesh

text: Cell-Centered

text: Scheme

text: for

text: One-Dimensional

text: Elastic-Plastic

text: Flows

label: author

text: Jun-Bo

text: Cheng

text: Weizhang

text: Huang

text: Song

text: Jiang

text: Baolin

text: Tian

text: A

label: abstract

text: third-order

text: moving

text: mesh

text: cell-centered

text: scheme

text: without

text: the

text: remapping

text: of

text: physical

text: variables

text: is

text: developed

text: for

text: the

text: numerical

text: solution

text: of

text: one-dimensional

text: elastic-plastic

text: ﬂows

text: with

text: the

text: Mie-Gru¨neisen

text: equation

text: of

text: state,

text: the

text: Wilkins

text: constitutive

text: model,

text: and

text: the

text: von

text: Mises

text: yielding

text: criterion.

text: The

text: scheme

text: combines

text: the

text: Lagrangian

text: method

text: with

text: the

text: MMPDE

text: moving

text: mesh

text: method

text: and

text: adaptively

text: moves

text: the

text: mesh

text: to

text: better

text: resolve

text: shock

text: and

text: other

text: types

text: of

text: waves

text: while

text: preventing

text: the

text: mesh

text: from

text: crossing

text: and

text: tangling.

text: It

text: can

text: be

text: viewed

text: as

text: a

text: direct

text: arbitrarily

text: Lagrangian-Eulerian

text: method

text: but

text: can

text: also

text: be

text: degenerated

text: to

text: a

text: purely

text: Lagrangian

text: scheme.

text: It

text: treats

text: the

text: relative

text: velocity

text: of

text: the

text: ﬂuid

text: with

text: respect

text: to

text: the

text: mesh

text: as

text: constant

text: in

text: time

text: between

text: time

text: steps,

text: which

text: allows

text: high-order

text: approximation

text: of

text: free

text: boundaries.

text: A

text: time

text: dependent

text: scaling

text: is

text: used

text: in

text: the

text: monitor

text: function

text: to

text: avoid

text: possible

text: sudden

text: movement

text: of

text: the

text: mesh

text: points

text: due

text: to

text: the

text: creation

text: or

text: diminishing

text: of

text: shock

text: and

text: rarefaction

text: waves

text: or

text: the

text: steepening

text: of

text: those

text: waves.

text: A

text: two-rarefaction

text: Riemann

text: solver

text: with

text: elastic

text: waves

text: is

text: employed

text: to

text: compute

text: the

text: Godunov

text: values

text: of

text: the

text: density,

text: pressure,

text: velocity,

text: and

text: deviatoric

text: stress

text: at

text: cell

text: interfaces.

text: Numerical

text: results

text: are

text: presented

text: for

text: three

text: examples.

text: The

text: third-order

text: convergence

text: of

text: the

text: scheme

text: and

text: its

text: ability

text: to

text: concentrate

text: mesh

text: points

text: around

text: shock

text: and

text: elastic

text: rarefaction

text: waves

text: are

text: demonstrated.

text: The

text: obtained

text: numerical

text: results

text: are

text: in

text: good

text: agreement

text: with

text: those

text: in

text: literature.

text: The

text: new

text: scheme

text: is

text: also

text: shown

text: to

text: be

text: more

text: accurate

text: in

text: resolving

text: shock

text: and

text: rarefaction

text: waves

text: than

text: an

text: existing

text: third-order

text: cell-centered

text: Lagrangian

text: scheme.

text: Key

text: Words.

text: Elastic-plastic

text: ﬂows;

text: Cell-centered

text: Lagrangian

text: scheme;

text: Moving

text: mesh

text: method;

text: High

text: order

text: scheme;

text: Hypoelastic

text: constitutive

text: model

text: Abbreviated

text: title.

text: A

text: third-order

text: moving

text: mesh

text: scheme

text: for

text: elastic-plastic

text: ﬂows

text: Laboratory

text: of

text: Computational

text: Physics,

text: Institute

text: of

text: Applied

text: Physics

text: and

text: Computational

text: Mathematics,

text: Beijing,

text: China

text: (cheng

text: junbo@iapcm.ac.cn)

text: Department

text: of

text: Mathematics,

text: the

text: University

text: of

text: Kansas,

text: Lawrence,

text: KS

text: U.S.A.

text: (whuang@ku.edu)

text: Laboratory

text: of

text: Computational

text: Physics,

text: Institute

text: of

text: Applied

text: Physics

text: and

text: Computational

text: Mathematics,

text: Beijing,

text: China

text: (jiang@iapcm.ac.cn)

text: Laboratory

text: of

text: Computational

text: Physics,

text: Institute

text: of

text: Applied

text: Physics

text: and

text: Computational

text: Mathematics,

text: Beijing,

text: China

text: (tian

text: baolin@iapcm.ac.cn)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: for this problem. The reference solution is obtained using the third order cell-centered Lagrangian scheme [?] with a uniform mesh of 4000 points. The MMCC scheme is applied to the problem up to t = 1. The L - and L -norms of the error at 1 2 the ﬁnal time t = 1 are shown in Tables ?? and ??. The results conﬁrm the anticipated third-order accuracy of the MMCC scheme. Table 1: The L -norm of the error for the MMCC scheme for the smooth solutions. 1 N ρ Order ρu Order ρE Order s Order xx 50 1.229E-03 - 1.680E-03 - 1.587E-03 - 7.685E-05 - 100 4.408E-04 1.48 4.260E-04 1.98 3.451E-04 2.20 2.540E-05 1.60 200 7.900E-05 2.48 7.430E-05 2.52 5.784E-05 2.58 4.241E-06 2.58 400 1.054E-05 2.91 9.737E-06 2.93 6.914E-06 3.06 4.366E-07 3.28 Table 2: The L -norm of the error for the MMCC scheme for the smooth solutions. 2 N ρ Order ρu Order ρE Order s Order xx 50 1.414E-03 - 2.187E-03 - 2.095E-03 - 8.911E-05 - 100 5.148E-04 1.46 5.852E-04 1.90 5.104E-04 2.04 3.276E-05 1.44 200 1.028E-04 2.32 1.164E-04 2.33 9.902E-05 2.37 6.068E-06 2.43 400 1.534E-05 2.74 1.681E-05 2.79 1.302E-05 2.93 7.097E-07 3.10 4.2 A piston problem with stress shock waves in copper This piston problem is concerned with a piece of copper with the following initial setting: the length of copper is 1m, the initial density ρ = 8930kg/m3, the initial pressure p = 105P a, and the 0 0 initial velocity is zero. The EOS for copper is given by the Mie-Gru¨neisen model with parameters ρ = 8930kg/m3, a = 3940m/s, Γ = 2, and s = 1.49. The constitutive model is characterized by the 0 0 × 0 × shear module µ = 45 109P a and the yield strength Y 0 = 90 106P a. A velocity boundary condition v = 20m/s is used on the left free boundary while the wall boundary condition is implemented piston on the right ﬁxed boundary. This problem has an exact solution; see [?]. To show the convergence of the MMCC scheme, we solve the problem with 100, 200, and 400 cells. The ﬁnal time is t = 150µs. The numerical results with diﬀerent N are shown in Figs. ?? and ??. From the ﬁgures we can see that the numerical solution is getting closer to the exact solution for larger N. Moreover, the scheme well captures the plastic shock wave (near x = 0.6) and the elastic shock wave (near x = 0.7) and there is no numerical oscillation near the shock waves. Furthermore, the yielding condition (7) is in eﬀect for 0 < x < 0.7 (cf. Fig. ??). It creates the plastic shock wave and also makes s to be only piecewise smooth. The L - norm of the error at the ﬁnal time t = 150µs xx 1 are shown in Table ??. Due to the discontinuity of the exact solution at the elastic shock wave and its 13 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: for

text: this

text: problem.

text: The

text: reference

text: solution

text: is

text: obtained

text: using

text: the

text: third

text: order

text: cell-centered

text: Lagrangian

text: scheme

text: with

text: a

text: uniform

text: mesh

text: of

text: points.

text: The

text: MMCC

text: scheme

text: is

text: applied

text: to

text: the

text: problem

text: up

text: to

text: t

text: The

text: L

text: and

text: L

text: -norms

text: of

text: the

text: error

text: at

text: the

text: ﬁnal

text: time

text: t

text: are

text: shown

text: in

text: Tables

text: and

text: The

text: results

text: conﬁrm

text: the

text: anticipated

text: third-order

text: accuracy

text: of

text: the

text: MMCC

text: scheme.

text: Table

text: The

label: caption

text: L

text: -norm

text: of

text: the

text: error

text: for

text: the

text: MMCC

text: scheme

text: for

text: the

text: smooth

text: solutions.

text: N

label: table

text: ρ

text: Order

text: ρu

text: Order

text: ρE

text: Order

text: s

text: Order

text: xx

text: Table

text: The

text: L

text: -norm

text: of

text: the

text: error

text: for

text: the

text: MMCC

text: scheme

text: for

text: the

text: smooth

text: solutions.

text: N

text: ρ

text: Order

text: ρu

text: Order

text: ρE

text: Order

text: s

text: Order

text: xx

text: A

text: piston

text: problem

text: with

text: stress

text: shock

text: waves

text: in

text: copper

text: This

text: piston

text: problem

text: is

text: concerned

text: with

text: a

text: piece

text: of

text: copper

text: with

text: the

text: following

text: initial

text: setting:

text: the

text: length

text: of

text: copper

text: is

text: the

text: initial

text: density

text: ρ

text: the

text: initial

text: pressure

text: p

text: a,

text: and

text: the

text: initial

text: velocity

text: is

text: zero.

text: The

text: EOS

text: for

text: copper

text: is

text: given

text: by

text: the

text: Mie-Gru¨neisen

text: model

text: with

text: parameters

text: ρ

text: a

text: Γ

text: and

text: s

text: The

text: constitutive

text: model

text: is

text: characterized

text: by

text: the

text: shear

text: module

text: µ

text: a

text: and

text: the

text: yield

text: strength

text: Y

text: a.

text: A

text: velocity

text: boundary

text: condition

text: v

text: is

text: used

text: on

text: the

text: left

text: free

text: boundary

text: while

text: the

text: wall

text: boundary

text: condition

text: is

text: implemented

text: piston

text: on

text: the

text: right

text: ﬁxed

text: boundary.

text: This

text: problem

text: has

text: an

text: exact

text: solution;

text: see

text: To

text: show

text: the

text: convergence

text: of

text: the

text: MMCC

text: scheme,

text: we

text: solve

text: the

text: problem

text: with

text: and

text: cells.

text: The

text: ﬁnal

text: time

text: is

text: t

text: The

text: numerical

text: results

text: with

text: diﬀerent

text: N

text: are

text: shown

text: in

text: Figs.

text: and

text: From

text: the

text: ﬁgures

text: we

text: can

text: see

text: that

text: the

text: numerical

text: solution

text: is

text: getting

text: closer

text: to

text: the

text: exact

text: solution

text: for

text: larger

text: N.

text: Moreover,

text: the

text: scheme

text: well

text: captures

text: the

text: plastic

text: shock

text: wave

text: (near

text: x

text: and

text: the

text: elastic

text: shock

text: wave

text: (near

text: x

text: and

text: there

text: is

text: no

text: numerical

text: oscillation

text: near

text: the

text: shock

text: waves.

text: Furthermore,

text: the

text: yielding

text: condition

text: is

text: in

text: eﬀect

text: for

text: x

text: (cf.

text: Fig.

text: It

text: creates

text: the

text: plastic

text: shock

text: wave

text: and

text: also

text: makes

text: s

text: to

text: be

text: only

text: piecewise

text: smooth.

text: The

text: L

text: norm

text: of

text: the

text: error

text: at

text: the

text: ﬁnal

text: time

text: t

text: xx

text: are

text: shown

text: in

text: Table

text: Due

text: to

text: the

text: discontinuity

text: of

text: the

text: exact

text: solution

text: at

text: the

text: elastic

text: shock

text: wave

text: and

text: its

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 0.6 100 cells 200 cells 400 cells 0.5 Exact 0.4 0.3 0.2 0.1 0 0.25 0.5 0.75 x 0 100 cells 200 cells 0.01 400 cells Exact 0.02 0.03 0.04 0.05 0.06 0.25 0.5 0.75 x Figure 3: Piston problem. The pressure and s at t = 150µs are shown. xx 16 )aPG( s )aPG(erusserP xx ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: cells

text: cells

text: cells

text: Exact

text: x

text: cells

text: cells

text: cells

text: Exact

text: x

text: Figure

text: Piston

text: problem.

text: The

text: pressure

text: and

text: s

text: at

text: t

text: are

text: shown.

text: xx

text: )aPG(

text: s

text: )aPG(erusserP

text: xx

text: ##LTFigure##

label: figure

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: MMCC, 200 cells CCL, 200 cells Reference 2950 2900 2850 2800 1 2 3 4 x 0.2 MMCC, 200 cells 0.15 CCL, 200 cells Reference 0.1 0.05 0 0.05 0.1 0.15 0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x Figure 10: Wilkins’ problem with the Mie-Gru¨neisen EOS. The density and s are shown for MMCC xx and CCL with 200 cells. 23 )3m/gk(ytisneD )aPG( s xx ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: MMCC,

text: cells

text: CCL,

text: cells

text: Reference

text: x

text: MMCC,

text: cells

text: CCL,

text: cells

text: Reference

text: x

text: Figure

text: Wilkins’

text: problem

text: with

text: the

text: Mie-Gru¨neisen

text: EOS.

text: The

text: density

text: and

text: s

text: are

text: shown

text: for

text: MMCC

text: xx

text: and

text: CCL

text: with

text: cells.

text: )aPG(

text: s

text: xx

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [29] V. T. Nguyen, An arbitrary Lagrangian-Eulerian discontinuous Galerkin method for simulations of ﬂows over variable geometries, J. Fluid Struct. 26 (2010), 312-329. [30] A. K. Pandare, C. Wang, and H. Luo, An arbitrary Lagrangian-Eulerian reconstructed Discon- tinuous Galerkin method for compressible multiphase ﬂows, 46th AIAA ﬂuid dynamic conference, June 2016, DOI:10.2514/6.2016-4270. [31] S. K. Sambasivan, R. Loubere and M. J. Shashkov, A ﬁnite volume Lagrangian cell-centered mimetic approach for computing elastoplastic deformation of solids in general unstructured grids, In: 6th European Congress on Computational Methods in Applied Sciences and Engineering 2012 (ECCOMAS 2012), Vienna, Austria, September 10-14, 2012. [32] T. Tang. Moving mesh methods for computational ﬂuid dynamics ﬂow and transport. In Recent Advances in Adaptive Computation (Hangzhou, 2004), volume 383 of AMS Contemporary Mathe- matics, pages 141–173. Amer. Math. Soc., Providence, RI, 2005. [33] E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer Dordrecht Heidelberg London New York, 1997. [34] J. A. Trangenstein and P. Collela, A high-order Godunov method for modeling ﬁnite deformation in elastic-plastic solids. Comm. Pure Appl. Math. XLIV (1991), 41-100. [35] M. L. Wilkins, Methods in Computational Physics, Vol.3, Chapter Calculation of elastic-plastic ﬂow, Academic Press, 1964, pp. 211-263. 27

label: reference

text: V.

text: T.

text: Nguyen,

text: An

text: arbitrary

text: Lagrangian-Eulerian

text: discontinuous

text: Galerkin

text: method

text: for

text: simulations

text: of

text: ﬂows

text: over

text: variable

text: geometries,

text: J.

text: Fluid

text: Struct.

text: A.

text: K.

text: Pandare,

text: C.

text: Wang,

text: and

text: H.

text: Luo,

text: An

text: arbitrary

text: Lagrangian-Eulerian

text: reconstructed

text: Discon-

text: tinuous

text: Galerkin

text: method

text: for

text: compressible

text: multiphase

text: ﬂows,

text: AIAA

text: ﬂuid

text: dynamic

text: conference,

text: June

text: S.

text: K.

text: Sambasivan,

text: R.

text: Loubere

text: and

text: M.

text: J.

text: Shashkov,

text: A

text: ﬁnite

text: volume

text: Lagrangian

text: cell-centered

text: mimetic

text: approach

text: for

text: computing

text: elastoplastic

text: deformation

text: of

text: solids

text: in

text: general

text: unstructured

text: grids,

text: In:

text: European

text: Congress

text: on

text: Computational

text: Methods

text: in

text: Applied

text: Sciences

text: and

text: Engineering

text: (ECCOMAS

text: Vienna,

text: Austria,

text: September

text: T.

text: Tang.

text: Moving

text: mesh

text: methods

text: for

text: computational

text: ﬂuid

text: dynamics

text: ﬂow

text: and

text: transport.

text: In

text: Recent

text: Advances

text: in

text: Adaptive

text: Computation

text: (Hangzhou,

text: volume

text: of

text: AMS

text: Contemporary

text: Mathe-

text: matics,

text: pages

text: Amer.

text: Math.

text: Soc.,

text: Providence,

text: RI,

text: E.

text: F.

text: Toro.

text: Riemann

text: Solvers

text: and

text: Numerical

text: Methods

text: for

text: Fluid

text: Dynamics,

text: Springer

text: Dordrecht

text: Heidelberg

text: London

text: New

text: York,

text: J.

text: A.

text: Trangenstein

text: and

text: P.

text: Collela,

text: A

text: high-order

text: Godunov

text: method

text: for

text: modeling

text: ﬁnite

text: deformation

text: in

text: elastic-plastic

text: solids.

text: Comm.

text: Pure

text: Appl.

text: Math.

text: XLIV

text: M.

text: L.

text: Wilkins,

text: Methods

text: in

text: Computational

text: Physics,

text: Chapter

text: Calculation

text: of

text: elastic-plastic

text: ﬂow,

text: Academic

text: Press,

text: pp.