A simple diffuse interface approach on adaptive Cartesian grids for the linear elastic wave equations with complex topography

Contents lists available atScienceDirect

Journal

of

Computational

Physics

www.elsevier.com/locate/jcp

A

simple

diffuse

interface

approach

on

adaptive

Cartesian

grids

for

the

linear

elastic

wave

equations

with

complex

topography

Maurizio Tavelli

a

,

Michael Dumbser

a

,

∗

,

Dominic

Etienne Charrier

b

,

Leonhard Rannabauer

c

,

Tobias Weinzierl

b

,

Michael Bader

c

a_{DepartmentofCivil,EnvironmentalandMechanicalEngineering,UniversityofTrento,ViaMesiano77,I-38123Trento,Italy} b_{DepartmentofComputerScience,UniversityofDurham,LowerMountjoy,SouthRoad,DurhamDH13LE,UnitedKingdom} c_{DepartmentofInformatics,TechnicalUniversityMunich(TUM),Boltzmannstr.3,D-85748Garching,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received25April2018

Receivedinrevisedform6February2019 Accepted7February2019

Availableonline18February2019 Keywords:

Diffuseinterfacemethod(DIM) Complexgeometries Highorderschemes

DiscontinuousGalerkinschemes Adaptivemeshrefinement(AMR) Linearelasticityequationsforseismicwave propagation

Inmost classicalapproaches of computational geophysicsfor seismic wavepropagation problems,complexsurfacetopographyiseitheraccountedforbyboundary-fitted unstruc-turedmeshes,or, where possible,by mappingthecomplex computational domainfrom physicalspacetoatopologicallysimple domaininareference coordinatesystem. How-ever, all these conventional approaches face problems if the geometry of the problem becomessufficientlycomplex.Theyeitherneedameshgeneratortocreateunstructured boundary-fittedgrids, whichcan become quite difficultand may require alot of man-ual user interactionsin orderto obtain ahigh quality mesh, orthey need the explicit computationofanappropriatemapping functionfromphysical toreference coordinates. ForsufficientlycomplexgeometriessuchmappingsmayeithernotexistortheirJacobian couldbecomeclosetosingular.Furthermore,inbothconventionalapproacheslowquality gridswillalwaysleadtoverysmalltimestepsduetothe Courant-Friedrichs-Lewy(CFL) conditionforexplicitschemes.Inthispaper, weproposeacompletelydifferentstrategy thatfollowstheideasofthesuccessfulfamilyofhighresolutionshock-capturingschemes, wherediscontinuities canactually beresolvedanywhereonthe grid, withouthaving to fitthemexactly.Weaddresstheproblemofgeometricallycomplexfreesurfaceboundary conditionsforseismicwavepropagationproblemswithanoveldiffuseinterfacemethod (DIM)on adaptive Cartesianmeshes (AMR)that consists inthe introduction ofa char-acteristicfunction0_≤

α

≤1 which identifiesthe locationofthe solidmediumand the surroundingair(or vacuum) andthus implicitlydefinesthe locationofthe freesurface boundary.Physically,

α

representsthevolumefraction ofthesolidmediumpresent ina controlvolume.Ournewapproachcompletelyavoids theproblemofmeshgeneration,since allthatisneededforthedefinition ofthecomplexsurfacetopographyistoset ascalar colorfunctiontounityinsidethe regionscoveredbythe solidandto zerooutside. The governingequations are derived from ideastypically used inthemathematical descrip-tionofcompressiblemultiphaseflows.AnanalysisoftheeigenvaluesofthePDEsystem showsthatthecomplexityofthegeometryhasnoinfluenceontheadmissibletimestep sizeduetotheCFLcondition.Themodelreducestotheclassicallinearelasticityequations insidethesolidmediumwherethegradientsof

α

arezero,whileinthediffuseinterface zoneatthefreesurfaceboundarythegoverningPDEsystembecomesnonlinear.We can provethatthesolutionoftheRiemannproblemwitharbitrarydataandajumpin

α

from

*

Correspondingauthor.

E-mailaddresses:m.tavelli@unitn.it(M. Tavelli),michael.dumbser@unitn.it(M. Dumbser),dominic.e.charrier@durham.ac.uk(D.E. Charrier), leo.rannabauer@tum.de(L. Rannabauer),tobias.weinzierl@durham.ac.uk(T. Weinzierl),bader@in.tum.de(M. Bader).

https://doi.org/10.1016/j.jcp.2019.02.004

0021-9991/©2019TheAuthor(s).PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

unitytozeroyieldsaGodunov-stateattheinterfacethatsatisfiesthefree-surface bound-aryconditionexactly,i.e. thenormalstresscomponentsvanish.Inthegeneralcaseofan interfacethatis notalignedwiththe gridandwhichisnot infinitelythin,asystematic studyonthedistributionofthevolumefractionfunctioninsidetheinterfaceandthe sen-sitivitywith respecttothethicknessofthe diffuseinterfacelayer hasbeencarried out. Inordertoreducenumericaldissipation,weusehighorderdiscontinuousGalerkin(DG) finiteelementschemesonadaptiveAMRgridstogetherwithasecondorderaccuratehigh resolutionshockcapturingsubcellfinitevolume(FV)limiterinthediffuseinterfaceregion. WefurthermoreemployalittledissipativeHLLEMRiemannsolver,whichisabletoresolve thesteadycontactdiscontinuityassociatedwiththevolumefractionfunctionandthe spa-tiallyvariablematerialparametersexactly.Whilethemethodislocallyhighorderaccurate intheregionswithoutlimiter,theglobalorderofaccuracyoftheschemeisatmosttwoif thelimiterisactivated.Itislocallyoforderoneinsidethediffuseinterfaceregion,which istypicalforshock-capturingschemes atshocks andcontact discontinuities.We showa large setofcomputationalresultsintwo andthreespacedimensionsinvolvingcomplex geometrieswherethe physicalinterfaceisnot alignedwiththegridorwhereitiseven moving.Foralltestcasesweprovideaquantitativecomparisonwithclassicalapproaches basedonboundary-fittedunstructuredmeshes.

©2019TheAuthor(s).PublishedbyElsevierInc.Thisisanopenaccessarticleunderthe CCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The numerical solution of linear elastic wave propagation is still a challenging task, especially when complex three-dimensionalgeometriesareinvolved.Inthepast,alargenumberofnumericalschemeshasbeenproposedforthesimulation ofseismicwavepropagation.Madariaga[1] andVirieux[2,3] introducedfinitedifferenceschemesforthesimulationof pres-sure(P)andshear(SVandSH)wavepropagation.Theseschemeswerethenextendedtohigherorder,see[4],threespace dimensions[5,6] andtoanisotropicmaterial[7,8].Forfinitedifference-likemethodsonunstructuredmesheswerefertothe workofMagnieretal. [9] andKäser&Igel[10,11].Therearealsoseveralapplicationsinthecontextoffinitevolume(FV) schemes[12–17],which,however,werealllimitedtosecond orderofaccuracyinspaceandtime. Thefirstarbitraryhigh orderADERfinite volumescheme forseismicwave propagationwas introduced in[18]. Forrealapplicationsit iscrucial thatanumericalschemeisabletoproperlycapturecomplexsignalsoverlongdistancesandtimes.Incontrasttoclassical loworderschemes,highordermethodsinspaceandtime areabletobetterreproducethetimeevolutionofthesolution. Aquantitativeaccuracyanalysisofhighordernumericalschemesforlinearelasticity,basedonthemisfitcriteriadeveloped in[19,20], canbe found in[21,22]. Spectral finiteelement methods [23] were successfullyappliedto linearelastic wave propagationina well-knownseriesofpaperofKomatitschandcollaborators[24–28].ForChebyshevspectralmethodsfor wavepropagationwerefertotheworkofTessmeretal. [29,8] andIgel[30].Foralternativedevelopmentsintheframework ofstabilizedcontinuous finiteelements appliedto elasticandacousticwave propagationwerefer to[31–33].Apartfrom wave propagationinthemedium, alsotheproperrepresentationofcomplexsurfacetopographyisachallenging task.For thispurpose,severalhighordernumericalschemesonunstructuredmesheswereintroducedinthepast.Aseriesofexplicit high order discontinuous Galerkin (DG) schemes for elastic wave propagation on unstructured mesheswas proposed in [34–39],whiletheconceptofspace-timediscontinuousGalerkinschemes,originallyintroducedandanalyzedin[40–46] for computationalfluiddynamics(CFD),waslateralsoextendedtolinearelasticityin[47–49].Thespace-timeDGmethodused in[49] isbasedon the novelconcept of

staggered discontinuous

Galerkin finiteelement schemes,which was introduced forCFD problemsin [50–56]. Inany case, all previous methods requirea boundary-fitted mesh that properly represents thegeometryofthephysicalproblemtobesolved.Thegenerationofthismeshisingeneralahighly non-trivialtaskand usuallyrequirestheuseofexternalmeshgenerationtools.Moreover,themeshgenerationprocessinhighlycomplex geom-etrycanleadtoverysmallelements withbadaspectratio,so-called

sliver elements [

57–59].Thiswellknownproblemcan oftenbeavoided,butnotalways,seee.g.[60,61].Forexplicittimediscretization,sliverelementscanonlybetreatedatthe aidof

local time stepping (LTS),

see,forexample,[36,62–64],butcurrentlyonlyveryfewschemesusedinproductioncodes employedincomputational seismologysupporttime-accuratelocaltimestepping. Alternatively,implicitschemes like[49] requiretheintroductionofaproperpreconditionerinordertolimitthenumberofiterationsneededtosolvetheassociated linearalgebraicsystem.

The key idea of this paper is therefore to completely avoid the mesh generation problem associated with classical

approachesusedincomputational seismology.Thisisachievedby extendingthelinearelasticwave equationsviaa char-acteristic (color)function

α

,whichis nothingelse than thevolume fractionofthe solid medium,andwhich determines ifapoint x islocatedinside thesolid material(

α

(

x

)

=

1)oroutside(

α

(

x

)

=

0). Inthiswaythe scalarparameter

α

sim-plydeterminesthephysicalboundarythroughadiffuseinterfacezone,insteadofrequiringaboundary-fittedstructuredor unstructuredmesh.Withthisnewapproach,evenverycomplexgeometriescanbe easilyrepresentedonregular adaptive Cartesianmeshes,i.e.viatheuseofadaptivemeshrefinement(AMR).Furthermore,theintroductionofthenewparameter

theCFLcondition.Tobemoreprecise:theadmissibletime stepsizeofthenewapproachpresentedinthispaperdepends ofcourseon thechosen

mesh spacing of

theregular AMRgridandonthe

signal speeds in

thePDEsystem,butitdoes

not

explicitlydependonthe

mesh quality and

the

geometric complexity of

thecomputationaldomain,asitisthecaseformany otherapproachesincomputationalseismology.

Inthecontextoffinitedifferenceschemesforseismicwavepropagation,immersedboundarymethodsforthetreatment ofcomplexfree surface topologiescanbefound, forexample,in[65–68] andreferencestherein.However, theunderlying mathematical models usedthereare

different from

the one proposed inthispaper,which isbasedon adiffuseinterface approachthatisusedforthedescriptionofcompressiblemulti-phaseflowsincomputationalfluiddynamics.

Our newmethod isinspired by the work concerning the modeling andsimulation ofcompressible multiphaseflows, see [69] and[70–73]. Itcan alsobe interpretedasa specialcaseofthemoregeneralsymmetrichyperbolic and thermo-dynamically compatiblemodel of nonlinear hyperelasticity of Godunov & Romenski andcollaborators [74–80]. A diffuse interface approach,similar totheoneused inthispaper,hasalreadybeensuccessfullyappliedto nonlinearcompressible fluid-structureinteractionproblemsinaseriesofpapers[81–83],buttheemployednumericalmethodswereonlyloworder accurateinspaceandtimeandthereforenotsuitableforseismicwavepropagationproblems.Otherapplicationsofdiffuse interface methods forcompressiblemulti-phaseflows canbe foundin[84–86], but,to thebestofour knowledge,thisis thefirsttimethatadiffuseinterfaceapproachisderivedandvalidatedforlinearseismicwavepropagationincomplex ge-ometries.Withinthepresentpaper,weusehighorderaccurateADER-DGschemesonCartesianmesheswithadaptivemesh refinement (AMR). Thenumericalmethod hasalreadysuccessfullybeenapplied toother hyperbolic PDEsystems [87,78]. The use ofadaptivemesh refinementallows to increase theresolution locallywhereneeded, especially closeto thefree surfaceoratinternalmaterialboundaries.Toavoidspuriousoscillationsandtoenforcenonlinearstability,weuseasimple but very robust

a posteriori subcell

finite volume(FV) limiter [88]. Here,a

second order total

variation diminishing(TVD) finite volumeschemeisadoptedinthe limitedDGcells.Inorderto maintain accuracy,thesubgridofthelimiter isbya factor of2N

+

1 times finercompared tothegrid ofan unlimitedDG schemewithpolynomial approximationdegree N. Theideaofusingan

a posteriori approach

tolimithighorderschemeswasfirstproposedbyClain,DiotandLoubèrewithin the so-calledMulti-dimensionalOptimal OrderDetection (MOOD)paradigm in thecontext offinitevolume schemes, see [89,90] formoredetails.Finally,inournumericalschemewemakeuseoftheHLLEMRiemannsolverintroducedin[91,92], whichisabletoresolvethesteadycontactdiscontinuities associatedwiththespatiallyvariablematerialparameters

λ

and

μ

(theLaméconstants),themassdensity

ρ

andthevolumefraction

α

.Thenumericalresultspresentedlaterinthispaper show that theproposed methodologyseems tobe avalidalternativeto existingapproachesincomputational seismology that arebasedonboundary-fittedstructuredorunstructuredmeshes.Atthispointwewouldliketostressthattheuseof a secondordershockcapturingTVDfinitevolumeschemeinsidethediffuseinterfaceregionatthefreesurface boundary limits the

global order

of accuracy ofthe schemeto atmosttwo. Numerical experimentsfurther show that the method is locallyfirstorder accurateinside thediffuseinterface region,whichis well-knownfromshockcapturingfinite volume schemes inCFD,whichalsoreducetofirstorderofaccuracyatshocksandcontactdiscontinuitiesthatare notexactly re-solvedonthegrid,see[93].Nevertheless,the

unlimited ADER-DG

schemethatisusedinsidethesolidmediumandfarfrom thefreesurfaceboundaryis

locally high

orderaccurateandthusbeneficialconcerningphaseandamplitudeerrorsforwave propagationoverlongdistancesandtimes

inside the

solidmedium.Notethatthemanifolddescribingthefreesurfaceisof onedimensionlessthanthecomputationaldomain,hencemostcellscanactuallyusethehighorderaccurate

unlimited DG

schemeandonlyveryfewcellsrequiretheuseofthesecondorderaccuratesubcellfinitevolumelimiter.Inordertoreduce thenumericalerrorsinthediffuseinterfaceregionasfaraspossible,weproposetouseadaptivemeshrefinement(AMR) withtime-accuratelocaltimestepping(LTS)combinedwithasubcellFVlimiter,wherethesubgridisbyafactorof2N

+

1 timesfinerthanthegridoftheunlimitedADER-DGschemewithpolynomialapproximationdegree

N.

However,wewould liketoemphasizethatthemathematicalmodelproposedinthispaperis

not strictly

linked tothenumericalschemesthat areusedinthispaperforitssolution(ADER-DGwithAMRcoupledwithsubcellfinitevolumelimiter).Anystandardfinite difference schemeinsidethe solidtogether withanonlinearENO/WENOscheme atthe freesurface where

∇

α

=

0 could havebeenappliedequallywell.

Therestofthepaperisorganized asfollows:inSection2weintroducethegoverningPDEofthenewdiffuseinterface approachforlinearelasticity.Wealsoshowthecompatibilityofourmodelwiththefreesurfaceboundaryconditioninthe casewhere

α

jumpsfrom1 to0.InSection 3we brieflysummarizethehighorderADER-DGschemesusedinthispaper. InSection4weshownumericalresultsforalargesetoftestproblemsintwoandthreespacedimensions,alsoincludinga realistic3DscenariowithcomplexgeometrygivenbyrealDTMdata.Finally,inSection5wegivesomeconcludingremarks andan outlookonfuturework, whichwillconcernnonlinear large-strainelasto-plasticityanddynamic ruptureprocesses inmovingmediabasedonthetheoryofnonlinearhyperelasticityofGodunovandRomenski[74,94,77].

2. Mathematicalmodel

Theequationsoflinearelasticity[95] canbewrittenas

∂

∂

t

σ

xx

− (λ +

2

μ

)

∂

∂

xu

− λ

∂

∂

yv

− λ

∂

∂

zw

=

Sxx

,

∂

∂

t

σ

y y

− λ

∂

∂

xu

− (λ +

2

μ

)

∂

∂

yv

− λ

∂

∂

zw

=

Sy y

,

∂

∂

t

σ

zz

− λ

∂

∂

xu

− λ

∂

∂

yv

− (λ +

2

μ

)

∂

∂

zw

=

Szz

,

∂

∂

t

σ

xy

−

μ



∂

∂

xv

+

∂

∂

yu



=

Sxy

,

∂

∂

t

σ

yz

−

μ



∂

∂

zv

+

∂

∂

yw



=

Syz

,

∂

∂

t

σ

xz

−

μ



∂

∂

zu

+

∂

∂

xw



=

Sxz

,

∂

∂

t

(

ρ

u

)

−

∂

∂

x

σ

xx

−

∂

∂

y

σ

xy

−

∂

∂

z

σ

xz

=

ρ

Su

,

∂

∂

t

(

ρ

v

)

−

∂

∂

x

σ

xy

−

∂

∂

y

σ

y y

−

∂

∂

z

σ

yz

=

ρ

Sv

,

∂

∂

t

(

ρ

w

)

−

∂

∂

x

σ

xz

−

∂

∂

y

σ

yz

−

∂

∂

z

σ

zz

=

ρ

Sw

,

(1)

where

λ

and

μ

arethesocalledLaméconstantsand

ρ

isthemassdensity.Inmorecompactformtheabovesystemreads

∂

σ

∂

t

−

E

(λ,

μ

)

· ∇

v

=

Sσ

,

(2)

∂

ρ

v

∂

t

− ∇ ·

σ

=

ρ

Sv

,

(3)

where v

= (

u

,

v

,

w

)

isthevelocityfield,

ρ

isthematerialdensity, S ρ and S σ arevolumeorpointsources,

σ

isthe sym-metricstresstensor,andE

(λ,

μ

)

isthestiffnesstensorthatconnectsthestraintensor



kl tothestresstensor

σ

according

totheHookelaw

σ

=

E



.Thestresstensor

σ

isgivenby

σ

=

⎛

⎝

σ

σ

xxyx

σ

σ

xyy y

σ

σ

xzyz

σ

zx

σ

zy

σ

zz

⎞

⎠

(4)

with the symmetry

σ

i j

=

σ

ji. The normal stress components are

σ

xx

,

σ

y y and

σ

zz, while the shear stress is

repre-sented by

σ

xy

,

σ

yz and

σ

xz. The stress tensor

σ

can thus be written in terms of its six independent components

(

σ

xx

,

σ

y y

,

σ

zz

,

σ

xy

,

σ

yz

,

σ

xz

)

. In the following we propose a new model that follows the ideas used in the simulation of

compressiblemultiphaseflows[69–71,73].InordertoderivethemodelwestartfromaBaer-Nunziato-typesystemforthe descriptionofcompressiblemulti-phaseflows,whereforthesolidphase(index

s)

thepressuretermhasbeenappropriately replacedbythestresstensor

σ

s,andwheretheusualpressureandvelocityrelaxationsourcetermshavebeendropped:

∂

∂

t

(

α

s

ρ

s

)

+ ∇ · (

α

s

ρ

svs

)

=

0

,

∂

∂

t

(

α

s

ρ

svs

)

+ ∇ · (

α

s

ρ

svs

⊗

vs

+

α

s

σ

s

)

−

σ

I

∇

α

s

=

α

s

ρ

sSv,s

,

∂

∂

t

(

α

s

ρ

sEs

)

+ ∇ · (

α

s

ρ

sEsvs

+

α

s

σ

svs

)

−

σ

I

∇

α

s

·

vI

=

α

s

ρ

sSv,s

·

vs

,

∂

∂

t



α

g

ρ

g

+ ∇ ·



α

g

ρ

gvg

=

0

,

∂

∂

t



α

g

ρ

gvg

+ ∇ ·



α

g

ρ

gvg

⊗

vg

+

α

g

σ

g

−

σ

g

∇

α

g

=

α

g

ρ

gSv,g

,

∂

∂

t



α

g

ρ

gEg

+ ∇ ·



α

g

ρ

gEgvg

+

α

g

σ

gvg

−

σ

I

∇

α

g

·

vI

=

α

g

ρ

gSv,g

·

vg

,

∂

∂

t

α

s

+

vI

∇

α

s

=

0

.

(5)

Hereindex

s refers

tothesolidphaseandindex

g refers

tothegasphasesurroundingthesolid;

ρ

kisthemassdensityand

Ekisthespecifictotalenergyofphase

k,

vk isthephasevelocity,vI istheso-calledinterfacevelocityand

σ

I isthestress

tensorattheinterface,whichisageneralizationoftheinterfacepressureusedinstandardBNmodels.Wenowmakethe followingsimplifyingassumptions:

(i) Theinterfacebetweenthesolidandthegasismovingonlyatanegligiblespeed,hencewecanassumevI

=

0.

(ii) Compared to the original Baer-Nunziato model [69,96,86], all pressure and velocity relaxation source terms are ne-glected.

(iii) Themassdensityofthegasphaseismuchsmallerthantheoneofthesolidphase(

ρ

g



ρ

s),hencethetimeevolution

ofthegas phaseis notrelevantforour purposes.Therefore,all evolutionequationsrelatedto thegasphase canbe neglectedinthefollowing,similartotheapproachusedin[97–99] inthecontextofnon-hydrostaticfreesurfaceflow simulationsbasedonadiffuseinterface approach.Toeasenotation,theremaining index

s for

thesolidphasecan be dropped.

(iv) Weassumethedensity

ρ

softhesolidphasetobeconstantintime,sotherelatedmassconservationequationcanbe

neglected.

(v) Thestress tensorofthesolid canbedirectlycalculatedviaHooke’s law(2),soitisnotnecessarytoevolve thetotal energyconservationlawforthesolid.

(vi) Thenonlinear convectiveterm

α

s

ρ

svs

⊗

vs,whichisquadraticinthesolid velocity, canbeneglected, sincethesolid

velocityisassumedtobesmallinthelinearelasticitylimit.

(vii) Last but not least, the free surface boundarycondition atthe interface betweensolid andsurrounding gasleads to

σ

I

· ∇

α

s

=

0.

As aresultofthesesimplifyingassumptions,the

reduced governing

PDEsystemofthenewdiffuseinterface approachfor linearelasticityincomplexgeometryreads:

∂

σ

∂

t

−

E

(λ,

μ

)

· ∇

v

=

Sσ

,

(6)

∂

αρ

v

∂

t

− ∇ · (

ασ

)

=

αρ

Sv

,

(7)

∂

α

∂

t

=

0

.

(8)

Since

∂

t

ρ

=

0,thepreviousequationsarethenrewrittenas

∂

σ

∂

t

−

E

(λ,

μ

)

·

1

α

∇(

α

v

)

+

1

α

E

(λ,

μ

)

·

v

⊗ ∇

α

=

Sσ

,

(9)

∂

α

v

∂

t

−

α

ρ

∇ ·

σ

−

1

ρ

σ

· ∇

α

=

Sv

,

(10)

∂

α

∂

t

=

0

.

(11)

Furthermorethefollowingequationsforthematerialparametersareaddedtothesystem:

∂λ

∂

t

=

0

,

∂

μ

∂

t

=

0

,

∂

ρ

∂

t

=

0

.

(12)

The material parameters

λ,

μ

and

ρ

are assumed to be constant in time butnot in space, i.e.

λ

= λ(

x

),

μ

=

μ

(

x

)

and

ρ

=

ρ

(

x

)

,forwhichwewilluseahighorderpolynomial representationasfortheother variablesofthePDEsystem.The samediffuseinterfacemodel(6)-(8) canalsobeobtainedbycombiningthenonlinearhyperelasticityequationsofGodunov andRomenski[74,94,100] withthecompressiblemulti-phasemodelofRomenskietal.[75,101],assuming linearmaterial behaviorandneglectingnonlinearconvectiveterms.System (9)-(12) isthenrewritteninthefollowingform:

∂

Q

∂

t

+

B1

(

Q

)

∂

Q

∂

x

+

B2

(

Q

)

∂

Q

∂

y

+

B3

(

Q

)

∂

Q

∂

z

=

S

(

x

,

t

),

(13)

wherethethreematricesB1,B2andB3 arespecifiedinEqs. (15)-(17).ThevectorQ isgivenby

Q

=



σ

xx

,

σ

y y

,

σ

zz

,

σ

xy

,

σ

yz

,

σ

xz

,

α

u

,

α

v

,

α

w

, λ,

μ

,

ρ

,

α



_,

₍₁₄₎

B1

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0 0 0 0 0 0

−

_α1

(λ

+

2

μ

)

0 0 0 0 0 _α1

(λ

+

2

μ

)

u 0 0 0 0 0 0

−

_α1

λ

0 0 0 0 0 _α1

λ

u 0 0 0 0 0 0

−

_α1

λ

0 0 0 0 0 _α1

λ

u 0 0 0 0 0 0 0

−

_α1

μ

0 0 0 0 _α1

μ

v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

−

_α1

μ

0 0 0 _α1

μ

w

−

α ρ 0 0 0 0 0 0 0 0 0 0 0

−

ρ1

σ

xx 0 0 0

−

α_ρ 0 0 0 0 0 0 0 0

−

_ρ1

σ

xy 0 0 0 0 0

−

α_ρ 0 0 0 0 0 0

−

_ρ1

σ

xz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,

(15) B2

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0 0 0 0 0 0 0

−

_α1

λ

0 0 0 0 _α1

λ

v 0 0 0 0 0 0 0

−

_α1

(λ

+

2

μ

)

0 0 0 0 _α1

(λ

+

2

μ

)

v 0 0 0 0 0 0 0

−

_α1

λ

0 0 0 0 _α1

λ

v 0 0 0 0 0 0

−

_α1

μ

0 0 0 0 0 _α1

μ

u 0 0 0 0 0 0 0 0

−

_α1

μ

0 0 0 _α1

μ

w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

−

α_ρ 0 0 0 0 0 0 0 0

−

_ρ1

σ

xy 0

−

α_ρ 0 0 0 0 0 0 0 0 0 0

−

1_ρ

σ

y y 0 0 0 0

−

α_ρ 0 0 0 0 0 0 0

−

_ρ1

σ

yz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

,

(16) B3

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

0 0 0 0 0 0 0 0

−

_α1

λ

0 0 0 _α1

λ

w 0 0 0 0 0 0 0 0

−

_α1

λ

0 0 0 _α1

λ

w 0 0 0 0 0 0 0 0

−

_α1

(λ

+

2

μ

)

0 0 0 _α1

(λ

+

2

μ

)

w 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

−

_α1

μ

0 0 0 0 _α1

μ

v 0 0 0 0 0 0

−

_α1

μ

0 0 0 0 0 _α1

μ

u 0 0 0 0 0

−

α_ρ 0 0 0 0 0 0

−

_ρ1

σ

xz 0 0 0 0

−

α_ρ 0 0 0 0 0 0 0

−

_ρ1

σ

yz 0 0

−

α_ρ 0 0 0 0 0 0 0 0 0

−

_ρ1

σ

zz 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.

(17)

TheeigenvaluesassociatedwiththematrixB1 are

λ

1

= −

cp

,

λ

2,3

= −

cs

,

λ

4,5,6,7,8,9,10

=

0

,

λ

11,12

= +

cs

,

λ

13

= +

cp

,

(18) where cp

=



λ

+

2

μ

ρ

and cs

=

μ

ρ

(19)

are the p

−

ands

−

wave velocities,respectively. The matrixof righteigenvectors ofthe matrix B1 asdefined in (15) is

R

=

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

ρ

c2_p 0 0 0 0 0 0 0 0

−

σ

xx 0 0

ρ

c2p

ρ

(

c2_p

−

2c2_s

)

0 0 1 0 0 0 0 0 0 0 0

ρ

(

c2_p

−

2c2_s

)

ρ

(

c2_p

−

2c2_s

)

0 0 0 1 0 0 0 0 0 0 0

ρ

(

c2_p

−

2c2_s

)

0

ρ

c2_s 0 0 0 0 0 0 0

−

σ

xy 0

ρ

c2s 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0

ρ

c2_s 0 0 0 0 0 0

−

σ

xz

ρ

c2s 0 0 cp 0 0 0 0 0 0 0 0

α

u 0 0

−

cp 0 cs 0 0 0 0 0 0 0

α

v 0

−

cs 0 0 0 cs 0 0 0 0 0 0

α

w

−

cs 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

α

0 0 0

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

.

(20)

The expressions for the eigenvalues andeigenvectors of B2 and B3 are very similar andcan be obtainedfrom those of B1,sincethePDE systemisrotationallyinvariant.Forthisreason,we donot givetheir explicitexpressionshere. Wenow

wanttoshow thattheproposed modelsatisfiesthefree surfaceboundarycondition

σ

·

n

=

0 exactlywhenconsidering a Riemannproblemthatincludesajumpof

α

from

α

L

₌

_{1 to}

_α

R

₌

_0.

Forthis,considertheleftandrightstateofaRiemannprobleminthe

x-direction

givenby

QL

= (

σ

xxL

,

σ

y yL

,

σ

zzL

,

σ

xyL

,

σ

yzL

,

σ

xzL

,

uL

,

vL

,

wL

, λ,

μ

,

ρ

,

1

),

(21)

QR

= (

σ

xxR

,

σ

y yR

,

σ

zzR

,

σ

xyR

,

σ

yzR

,

σ

xzR

,

0

,

0

,

0

, λ,

μ

,

ρ

,

0

).

(22)

Byusingasimplestraightlinesegmentpath

ψ (

s

)

=

QL

+

s

(

QR

−

QL

) ,

(23)

wecandefineageneralizedRoe-averagedmatrixB

˜

1in

x direction

accordingto[102–104] asfollows:

˜

B1

=

1



0 B1

(ψ (

s

))

ds

.

(24)

TheexactsolutionofthelinearizedRiemannproblembasedontheRoe-averagedmatrixB

˜

1

= ˜

R

˜˜

R−1 aboveandthe

simi-laritycoordinate

ξ

=

x

/

t reads

QRP

(ξ )

=

1 2R

˜



I

−

sign

(ξ

I

− ˜)



R

˜

−1

·

QL

+

1 2R

˜



I

+

sign

(ξ

I

− ˜)



R

˜

−1

·

QR

,

(25)

with I beingthe identity matrix.From QRP

(ξ )

we can obtainthe following Godunov state QGod

=

QRP

(

0

)

at theinterface

(

ξ

=

0) QGod

=



0

,

σ

L xxc2p

+

2

σ

xxLc2s

+

σ

y yc2p c2p

,

σ

L xxc2p

+

2

σ

xxLc2s

+

σ

zzc2p c2p

,

0

,

σ

_yzL

,

0

,

cp

ρ

uL

−

σ

xxL cp

ρ

,

cs

ρ

v L

₋

_σ

L xy cs

ρ

,

cs

ρ

w L

₋

_σ

L xz cs

ρ

, λ,

μ

,

ρ

,

1



,

fromwhichitisclearthatallthecomponentsofthenormalstressin

x-direction

(

σ

xx

,

σ

xy and

σ

xz)arezero,whichmeans

that thefreesurface boundarycondition

σ

·

n

=

0 isindeedrespected bymerelyimposing ajumpinthevolumefraction functionfrom

α

=

1 to

α

=

0.

Asonecannote,themodel(9)-(11) involvesdivisionsby

α

thatcanbeasourceofinstabilitiesattheinterface,sincethe colorfunction

α

isideallysettozero,oratleastclosetozero,outsidethesolidmedium.Inordertoaddressthisproblem, we introduceasimpletransformationthat avoidsthedivisionsbyzeros.Inparticular,wesubstituteall multiplicationsby

α

−1

₌

₁

_/

_α

_,with

α

−1

∼

=

α

α

2

₊

_

₍

_α

₎

,

(26)

where



=



(

α

)

hastosatisfy



(

1

)

=

0 and



(

0

)

=



0

>

0 inordertobe consistentwiththelinearelasticityequations.In

ourcasewe takeasimplelinearfunction



=



0

(

1

−

α

)

with



0

=

10−3.Theintroductionofthisnewparameterwiththis

methodismandatorytoobtainastablesolution.Theneweigenvaluesare

˜λ =

f

λ

,where f

=

 α

α2_+0(1−α) thatfor

α

∈ [

0

,

1

]

satisfies f

∈ [

0

,

1

]

and f

=

1 for

α

=

1.

Assoonasweuseanon-trivialgeometrywechooseadiffuseinterfaceof

finite width I

D forthetransitionbetweenthe

solid medium

α

=

1 and the surroundinggas / vacuum(

α

=

0). For arelatively large width ID ofthe diffuseinterface,

therearesomequestionsthatarisenaturallyconcerningthedistributionofthecharacteristicfunction

α

insidethediffuse interface andtheresultingeffectivepositionofthefree surfaceboundary. Ingeneral,it isimportanttosetupthe diffuse interfaceshapesuchthat

∇

α

isorientedasthenormalvectortothephysicalsurface,i.e.

∇

α

≈

n.Asimplewaytodothis istorepresentthetransitionregionbyapiecewisepolynomial.Let

r

=

r

(

x

)

bethesigneddistancebetweentherealphysical interfacelocationandagenericpointx under consideration.Wethendefine theshape ofthediffuseinterface asfunction ofafiniteinterfacethickness

I

D

≥

0,ashiftingparameter

η

andtheauxiliaryfunction:

ξ(

r

)

=

⎧

⎨

⎩

1 if r

>

(

1

+

η

)

ID

,

0 if r

<

−(

1

−

η

)

ID

,

r+(1−η)ID 2ID if r

∈ [−(

1

−

η

)

ID

, (

1

+

η

)

ID

].

(27) Wefinallydefinethesolidvolumefractionas

α

(

r

)

= (

1

− ξ(

r

))

p

,

(28)

where p

>

0 isan exponentthat determines theshape ofthe diffuseinterface. The widthofthe interface ID should be

relatedtothelocalsize

h of

thecomputationalmesh,i.e.onewouldtypicallychoose

I

D

∼

h. Inordertoreduce

I

D asmuch

aspossible,wewillmakeuseofadaptivemeshrefinement (AMR)incombinationwitha subcellfinitevolumelimiter, as discussedinthenextsection.

3. Numericalscheme

ThenumericalmethodthatweuseinordertosolvethePDEsystemintroduced inthepreviousSection2isan explicit ADER-DGschemeofarbitraryhighorderofaccuracyinspaceandtimeonadaptiveCartesiangrids (AMR).Thenumerical methodwas presentedfordifferentPDE systemsin[88,87,78],henceinthe followingweonlygive abriefsummary.The PDEsystem(9)-(11) canbewrittenincompactmatrix-vectornotationas

∂

Q

∂

t

+

B

(

Q

)

· ∇

Q

=

S

(

x

,

t

),

(29)

where Q is the state vector, B

(

Q

)

· ∇

Q is a non conservative product (see [105,102,103])and S

(

x

,

t

)

is a known source term.Inregions where

α

=

1 andthus

∇

α

=

0,thePDEsystem(29) reducestotheclassicallinearelasticwaveequations (1), while for

∇

α

=

0 the system becomes locally nonlinear and therefore requires a very robust numerical scheme as well ashighresolution tobeproperly solved.Withinthispaperweusethesimple andvery robustsubcellfinite-volume limiterapproachincombinationwithadaptivemeshrefinement(AMR).Adetaileddescriptionofthelimitercan befound in [88,87]. As suggested in [88], we employ Ns

= (

2N

+

1

)

d subgrid cells for the finite volume limiter, where d is the

numberofspacedimensionsoftheproblemand

N is

thepolynomialapproximationdegreeusedinthehighorderADER-DG scheme.Notethattheuseofsuchafinesubgridwithinthesubcellfinitevolumelimiterdoes

not reduce

thetimestepof the overall scheme, since finitevolume schemes are stableup to CFL

=

1, while DGschemes require CFL

<

1

/(

2N

+

1

)

. The

d-dimensional

computationaldomain



isdiscretized withanadaptive Cartesiangrid composedofCartesian control volumes

T

iinspaceas



=

Ne



i=1 Ti

,

(30)

where Ne isthe total number ofelements. Since we are interested ina high orderscheme, we first define a piecewise

polynomialnodalbasis

{φ

k

}

k=1...(N+1)d asthesetofLagrange polynomialspassing throughtheGauss-Legendrequadrature

pointsona referenceunitelement

T

re f foragivenpolynomial degree N

≥

0 anddimension

d.

Aweakformulationofthe

PDEsystem isobtainedaftermultiplying Eq.(29) by atest function

φ

k for

k

=

1

. . . (

N

+

1

)

d andthen integratingovera

space-timecontrolvolume Ti

× [

tn

,

tn+1

]

: tn+1



tn



Ti

φ

k



∂

Q

∂

t

+

B

(

Q

)

· ∇

Q



dx dt

=

tn+1



tn



Ti

φ

kS

(

x

,

t

)

dx dt

.

(31)

Werestrictthediscretesolutiontothespaceofpiecewisepolynomialsofdegree

N,

i.e. thenumericalsolutionuhiswritten

insideeachelement

T

i intermsofthepolynomialbasisas

uh

(

x

,

tn

)



_Ti

=

(N+1)d



k=1

forx

∈

Tiand

i

=

1

. . .

Ne.Thevectorofdegreesoffreedomofuh

(

x

,

tn

)

isdenotedbyu

ˆ

ni.Throughoutthepaperweusethe

Einstein summationconventionoverrepeatedindices.Usingthedefinition(32) in theweakformulationgivenby Eq.(31) weobtain(seealso[106,107])

⎛

⎜

⎝



Ti

φ

k

φ

ldx

⎞

⎟

⎠



u

ˆ

n_l,+_i1

− ˆ

un_l,i



+

tn+1



tn



∂Ti

φ

k

D

−

(

qh−

,

q+h

)

·

n dS dt

+

tn+1



tn



T_i◦

φ

kB

(

qh

)

· ∇

qhdx dt

=

tn+1



tn



Ti

φ

kS

(

x

,

t

)

dx dt

,

(33) wherewehaveintroducedthejumpcontribution

D

−

(

q−_h

,

q_h+

)

·

n ontheelementboundariesandthespace-timepredictor solution qh

(

x

,

t

)

.Moredetails concerningthecomputation ofqh

(

x

,

t

)

willbe reportedlater.Fortheapproximation ofthe

jumpterm

D

− weuseapath-conservativeschemeasintroducedbyParésin[103] andCastroetal. in[102].Weintroduce aLipschitzcontinuouspathfunction

ψ(

q−_h

,

q+_h

,

s

)

definedfor

s

∈ [

0

,

1

]

suchthat

ψ(

q_h−

,

q+_h

,

0

)

=

q−_h and

ψ(

q−_h

,

q+_h

,

1

)

=

q+_h, where q_h− denotes the boundary-extrapolated state fromwithin the element Ti and q+_h theboundary-extrapolated state

from the neighbor element. The simplestpossible choice for

ψ

, whichwe usein thispaper, is thelinear segment path betweenthetwostatesq−_h andq+_h:

ψ (

q−_h

,

q+_h

,

s

)

=

q_h−

+

s



q_h+

−

q−_h

.

(34) Following[105,103,102] wenowdefine thejumpcontribution

D

−

(

q−_h

,

q+_h

)

·

n sothatitsatisfiesthegeneralized Rankine-Hugoniotconditions

D

−

(

q−_h

,

q+_h

)

·

n

+

D

+

(

q−_h

,

q+_h

)

·

n

=

1



0 B

(ψ (

q−_h

,

q+_h

,

s

))

·

n

∂ψ

∂

sds

.

(35)

Thepreviouspathintegralcansimplybeevaluated

numerically using

asufficientnumberofGaussianquadraturepoints.As RiemannsolverweusethenewHLLEM-typeRiemannsolverfornon-conservativesystemsrecentlydescribedin[92],since wewanttoexactlypreservethecontactdiscontinuitiesofthematerialparametersandofthevolumefractionfunctionthat appearinthePDEsystem.Atthispointwewouldliketoemphasizethatinthispaperwedeliberatelyusefinitevolumeand discontinuous Galerkinfiniteelement schemesthat employapiecewise polynomial approximationspace, whichexplicitly permits jumps in the discrete solution atelement interfaces.In thiscontext the path-conservative schemes can properly deal withjumps in

α

if they are

exactly resolved

atan element interface, thusnaturally allowing todiscretize problems alsowith ID

=

0,seealso[102,103] foradiscussion inthecontextofshallowwaterequationswithdiscontinuous bottom

topography.Inthecaseof

I

D

=

0,ourmethodbecomesagainasharpinterfacemethodifthejumpsareexactlyresolvedon

thegrid,butthisisactually

not the

mainobjectiveofthepresentpaper.

Regarding the space-time predictor, we need to introduce a new polynomial basis of degree N in space and time

{θ

k

}

k=1...(N+1)d+1 wherenow

θ

k

(

x

,

t

)

contains alsothe time. We represent qh

(

x

,

t

)

in termsof thisnew space-timebasis

as qh

(

x

,

t

)

=

(N+1)d+1



k=1

θ

k

(

x

,

t

)

q

ˆ

nk

.

(36)

Let

T

◦_i

=

Ti

− ∂

Ti denotetheinteriorof

T

i andT_ist

=

T_i◦

× [

tn

,

tn+1

]

denotethenewspace-timecontrolvolume.The

space-timepredictoristhencomputedasanelement-localsolutionofthefollowingweakformulationofthePDEsystem(29):



Tst i

θ

k

∂

qh

∂

t dx dt

+



Tst i

θ

kB

(

qh

)

· ∇

qhdx dt

=



Tst i

θ

kS

(

x

,

t

)

dx dt

,

(37)

for

k

=

1

. . . (

N

+

1

)

d+1.UsingintegrationbypartsinthefirsttermofEq.(37) weobtaintwospatialcontributionson

T

iat

tn+1 and

t

n _{andan internalone since}

_θ

k

= θ

k

(

x

,

t

)

contains explicitlythetime.Forthespatial contributionattime

t

n we

use thenumericalsolution fromtheprevious time step.Notice that thiscorresponds toupwindingin thetime direction duetothecausalityprinciple.OnethusobtainsthefollowingweakformulationofthePDE

in the small [

108]:



Ti

θ

k

(

x

,

tn+1

)

qh

(

x

,

tn+1

)

dx

−



Ti

θ

k

(

x

,

tn

)

uh

(

x

,

tn

)

dx

−



T_ist

∂θ

k

∂

t qh

(

x

,

t

)

dx dt

+



T_ist

θ

kB

(

qh

)

· ∇

qhdx dt

=



T_ist

θ

kS

(

x

,

t

)

dx dt

.

(38)

SinceEq.(38) iselement-localitcanbesolvedusingasimpleandefficientPicardmethodwithoutanycommunicationwith theneighborelements,seee.g.Dumbseretal. [109].

Thenumericalschemeisconstrainedbya

local CFL-type

stabilitycondition,see[109,110,87],thatisgivenby



t

<

CFL d h 2N

+

1 1

|λ

max

|

,

(39)

where

h is

thelocal meshsize,

λ

max isthe maximumeigenvalueofthe PDEsystem, andCFL

<

1 is theCourantnumber,

which should be chosen according to [109] in order to have linear stability. Concerning the adaptive mesh refinement (AMR) werely onthe ExaHyPEenginehttp://exahype.eu,which isbuiltupon thespace-tree implementationPeano [111,

112] realizing cell-by-cellrefinement [113]. Forfurther details aboutAMR incombination withhighorder finitevolume andDGschemeswithtime-accuratelocaltimestepping(LTS)usedinthiswork,see[110,114–116].Throughoutthispaper, weuseadaptivemesheswitharefinementfactorof

r

=

3 betweentwoadjacentlevelsofrefinement.Inordertoillustrate theusefulcombinationofAMR withthe subcellfinitevolumelimiter, weprovidethe followingexample:withtwo levels ofmeshrefinementandfora DGschemewithpolynomial approximationdegree N

=

3,thefinitevolume subgridonthe finestAMR levelwillbeby afactorof

r

2

· (

2N

+

1

)

=

32

· (

2

·

3

+

1

)

=

63 finerthanthegrid usedfortheDGschemeon the coarsestgrid level.Thiscorresponds to amesh refinementof almosttwo orders ofmagnitude. However, thishasno negative impactatall onthetime stepsizeusedonthecoarsestgridlevel,thankstotheuseoftime-accuratelocaltime stepping(LTS),whichisstraightforwardforADERdiscontinuousGalerkinandADERfinitevolumeschemes,see[36,110,87]. ThecombinationofAMRwiththefinesubgridusedforthefinitevolumelimiterisnecessarytoalleviatethelossofformal orderofaccuracyinsidethelimitedcellsandinordertoallowasmallinterfacewidthID

∼

h.

Inordertodecidewheretorefine,weintroduceasimplerefinementindicatorfunctionnamed

ϕ

=

ϕ

(

x

,

t

)

thatdefines theobservedvariablefortherefinement/recoarsening processanda socalledreal-valued

estimator function

χ

=

χ

[

ϕ

]

,see again[110] formoredetails.Afterdefiningtheindicatorfunction,wedefinethecell-averagesof

ϕ

as

ˆ

ϕ

i

=

1

|

Ti

|



Ti

ϕ

(

x

,

t

)

dx

∀

i

=

1

. . .

Ne

,

(40)

andthenwecomputetheestimatorfunctionas

χ

i

[

ϕ

] =

max c∈Vi

 ˆ

ϕ

c

− ˆ

ϕ

i



/



xc

−

xi



,

(41)

where

V

i containsalltheVoronoineighborelementsof

T

i.Ourestimatorfunction

χ

issimplybasedonanapproximation

ofthe gradientofthe solutionin severalspatial directions[110]. Withtheseingredients athand,we introducea simple rulefortherefinement/recoarseningprocessbasedontwothresholds

χ

+and

χ

− asfollows:

(i) if

χ

i

[

ϕ

]

>

χ

+ then

T

i islabeledformeshrefinement;

(ii) if

χ

[

ϕ

]

<

χ

−then

T

iislabeledformeshrecoarsening.

Withinthispaper,wealwaysuse

ϕ

(

x

,

t

)

=

ϕ

(

Q

)

=

α

,

χ

+

=

0

.

01 and

χ

−

=

0

.

001.Wewillalsousethevolumefraction

α

to specifythezoneswheretoactivatethesubcellfinitevolumelimiter[88].Inparticular,weactivatetheFVlimiterwhenever

α

∈ [

/



,

1

−



]

,with



=

10−3.Aslongasthetopologyofthegeometrydescribedby

α

issupposedtobestationaryintime, wecanconsidertherefinementandthelimitedzonesalsoassteadyandthereforetheyneedtobeidentifiedonlyoncein themeshinitializationstep.

We stress againthat theuseof atime-accurate localtime stepping strategy (LTS)is mandatory inorderto avoidthe reductionofthetimestepsize



t in regionsthatarefarawayfromthediffuseinterface.

4. Numericalresults

4.1. Reflected plane wave

The purposeofthisfirsttest problemisto systematically studytheinfluenceof thewidth ID of thediffuseinterface

layerontothenumericalresults.Wealsoshowthatthemodelindeedconvergestothecorrectsolutioninthelimit ID

→

0.

We take a simple plane wave impulse in a domain



= [−

1

,

1

]

× [−

0

.

1

,

0

.

1

]

initially placed at x0

= −

0

.

25 andhitting

a free surface boundary placed in xD

=

0. The Laméconstants are chosen as

λ

=

2,

μ

=

1 and

ρ

=

1. We define Q0

=

(

0

,

0

,

0

,

0

,

0

,

0

,

0

,

0

,

0

,

λ,

μ

,

ρ

,

α

(

x

))

and

δ

= (

0

.

4

,

0

.

2

,

0

.

2

,

0

,

0

,

0

,

−

0

.

2

,

0

,

0

,

0

,

0

,

0

,

0

)

andset

Q

(

x

,

y

,

t

=

0

)

=

Q0

+ δ ·

e− (x−x0)2

2

_,

withthehalfwidth



=

0

.

05.Thevolumefractionfunction

α

(

x

)

isprescribedaccordingto(28) and(27).WeuseanADER-DG P4 schemeanda uniform Cartesian gridwith 100

×

2 elements. The mesh resolutionis chosen fine enoughso that the

Fig. 1. Numericalresultsobtainedwiththenewdiffuseinterfaceapproachforaplanewavereflectionproblemonafreesurfacelocatedinx=0 usinga variableinterfacethicknessofID=0,ID=0.001,ID=0.01 andID=0.03.Inallfourcaseswereportthevelocitycomponentu comparedwiththeexact solutionoftheproblem(bottom)togetherwiththespatialdistributionof

α

(top).

numerical results are grid-independent and only depend on the choice of the interface thickness ID. Since for this test

cp

=

2,theexactsolutionattime

t

=

tend

=

0

.

25 isthereflected p-wavewhichislocatedagainintheinitialposition.We

considerfourcaseswithdifferentchoicesoftheinterfacewidth ID,rangingfrom

I

D

=

0

.

03 tothelimit ID

=

0,wherethe

interfaceisexactlylocatedonacellboundary.FromtheresultsdepictedinFig.1wecanconcludethatthediffuseinterface methodisabletoreproducetheexactsolutionoftheproblemforsufficientlysmallvaluesoftheinterfacethickness

I

D.We

alsostressthattheuseofapath-conservativemethodallowsustoreduce theinterfacethicknessexactlyto

I

D

=

0,which

leads toajumpin

α

atan elementinterface,butwhichisstillproperlyaccountedforthankstothejumpterms

D

− used inthenumericalscheme.

Forratherlargevaluesofthefiniteinterfacethickness

I

D,wheretheactualshapeofthespatialdistributionof

α

startsto

playarole,wehavefoundempiricallythatagoodchoicefortheparameters

η

andp in (27) is

η

= −

0

.

6 and p

=

0

.

5.This choice allows toobtainstill acorrectreflectionofap-waveeven forvery thick interfaces.However, forsufficientlysmall valuesof

I

D,thechoiceof

η

and p has onlyverylittleinfluence.Weconsidernowasimilarsetupofap-wavetravelingin

aheterogeneousmaterialwithperiodicboundarieseverywhere.TheLaméconstantsarespecifiedas

(λ,

μ

,

ρ

)

=



(

4

,

0

,

1

)

0

≤

x

≤

0

.

2

(

2

,

1

,

1

)

otherwise (42)

Note that inthearea 0

≤

x

≤

0

.

2 thePoissonratiois

ν

=

1

/

2.The abilityofADER-DG schemesto dealwithfluids(

μ

=

0) properlyhas alreadybeendiscussed in [34,117]. We furthermoreplace a reflectivefree surface atxf s

=

0

.

75 only by

using a change inthe parameter

α

sothat the exact solution attend

=

1

.

0 isagain thereflected p-wave, located in the

initial position.We usean ADER-DGP3 schemeona sequenceofuniformCartesian grids inorderto checkthe accuracy

of the scheme.In Table 1 we report the L2-error norm inthe case whereno free surface boundary appears (i.e.

α

=

1

everywhere). In thiscasethewave passesthroughthe heterogeneousmaterial andreturnstothe initialstate dueto the periodicboundary.WeobservehighorderofconvergenceifweuseapureDGschemewithoutlimiter(leftcolumn),while we observetheexpecteddecayoftheorderifweartificiallyactivatethelimitercloseto x

=

xf s.Taking

α

=

0 for

x

≥

xf s,

we observeafirstorder ofconvergence,since thejumpin

α

alsorepresentsajumpinthe discretesolution,seeTable2. Thisisobservedalsoforpositivevaluesof

I

D anddifferentvaluesof



0,asreportedintherightcolumnofTable2.

InprincipleonecankeeptheLaméparametersinsidethediffuseinterfaceandoutsidethesolidastheonesinsidethe solid.HoweverwehaveempiricallyfoundoutthatrescalingtheLaméconstantsaccordingtothelocalvalueof

α

improves thenumericalconvergenceandallowstoreachsecondorderofaccuracy,seeTable3.

Table 1

ComputedL2-errornormforthecase

α

=1 withoutthelimiter(left)and

limitingtheelementsthatareclosetoxf s(right). Elements L2(u) 20×2 4.8988e-3 − 40×2 7.2321e-4 2.8 80×2 4.1602e-5 4.1 160×2 2.4629e-6 4.1 Elements L2(u) 20×2 6.6547e-3 − 40×2 1.3748e-3 2.3 80×2 2.1391e-4 2.7 160×2 3.7588e-5 2.5 Table 2

ComputedL2-errornormforthecaseof

α

alignedwiththemesh(left)andforpositivediffuseinterfacesizeID(right). Elements L2(u) 320×2 2.5851e-4 − 640×2 1.2104e-4 1.1 1280×2 4.6230e-5 1.4 2560×2 2.0770e-5 1.2 Elements ID L2(u),0=10−3 L2(u),0=10−4 40×2 0.02 3.4813e-3 − 3.2060e-3 − 80×2 0.01 2.0510e-3 0.8 2.0144e-3 0.7 160×2 0.005 1.1723e-3 0.8 1.1374e-3 0.8 320×2 0.0025 5.9087e-4 1.0 5.9268e-4 0.9 Table 3

Computed L2-error norm for α

aligned with the mesh and rescaledLaméconstantsλandμ.

Elements L2(u) 40×2 1.4512e-2 − 80×2 2.0014e-4 2.9 160×2 3.2374e-5 2.6 320×2 8.9161e-6 1.9

Fig. 2. Setupofthescatteringtestproblem.AMRgridanddistributionofthecharacteristicfunction

α

(left).Detailofthefreesurfacelocation∂C shown viaadashedlineand

α

colorcontours(center).Limitedcellshighlightedinredandunlimitedcellsshowninblue(right).(Forinterpretationofthecolors inthefigure(s),thereaderisreferredtothewebversionofthisarticle.)

4.2. Scattering of a plane wave on a circular cavity

In thistest casewe consideran initially planar p-wave travelingin x-direction inside a solid medium andhitting an emptycircularcavity.Theinitialstateisgivenby

Q

(

x

,

0

)

= (

0

,

0

,

0

,

0

,

0

,

0

,

0

,

0

,

0

, λ,

μ

,

ρ

,

α

)

+

0

.

1

· (

4

,

2

,

2

,

0

,

0

,

0

,

−

2

,

0

,

0

,

0

,

0

,

0

,

0

)

sin

(

2

π

x

),

(43) with

λ

=

2,

μ

=

1 and

ρ

=

1.The value of

α

is parameterized throughthecircularsurface C

= {(

x

,

y

)

|

x

2

₊

_y2

_≤

₀

_.

₂₅2

_}

sothat

α

(

x

)

=

0 if x

∈

C and α

=

1 if x

∈

/

C . Thewidthparameterofthe diffuseinterface issetto ID

=

0

.

01 on

∂

C . The

computational domainis



= [−

3

,

3

]

2 andtheinitial Cartesian gridatlevel



=

0 consistsof80

×

80 cells.We then use onefurtherrefinementlevel



max

=

1 basedonthegradientof

α

inordertorefinethemeshclosetothediffuseinterface.

Furthermore,we usea fifth orderADER-DG methodbased onpiecewise polynomials ofdegree N

=

4 in both spaceand time,supplementedwithasecondorderTVDsubcellfinitevolume limiter.TheresultingAMRgridandthecolorcontours of

α

areshowninFig.2,togetherwiththeregionwherethesubcellfinitevolumelimiterisactivated.Fromtheplotinthe centralpanel ofFig.2onecan seethatthewidthoftheinterface layerisoftheorderofthesizeofonecell ofthe high orderDGscheme.