Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics.

Transcription

1 Multi-sale Gounov-type metho for ell-entere isrete Lagrangian hyroynamis. Pierre-Henri Maire, Bonifae Nkonga To ite this version: Pierre-Henri Maire, Bonifae Nkonga. Multi-sale Gounov-type metho for ell-entere isrete Lagrangian hyroynamis.. Journal of Computational Physis, Elsevier, 2009, Volume 228 (Issue 3, 20 February 2009), pp.pages <0.06/j.jp >. <inria > HAL I: inria Submitte on 26 Jun 2008 HAL is a multi-isiplinary open aess arhive for the eposit an issemination of sientifi researh ouments, whether they are publishe or not. The ouments may ome from teahing an researh institutions in Frane or abroa, or from publi or private researh enters. L arhive ouverte pluriisiplinaire HAL, est estinée au épôt et à la iffusion e ouments sientifiques e niveau reherhe, publiés ou non, émanant es établissements enseignement et e reherhe français ou étrangers, es laboratoires publis ou privés.

2 Multi-sale Gounov-type metho for ell-entere isrete Lagrangian hyroynamis. Pierre-Henri Maire an Bonifae Nkonga UMR CELIA, Université Boreaux I, Talene Ceex, Frane IMB, Université Boreaux I, INRIA Boreaux-Su Ouest, Talene Ceex, Frane June 26, 2008 Abstrat This work presents a multiimensional ell-entere unstruture finite volume sheme for the solution of multimaterial ompressible flui flows written in the Lagrangian formalism. This formulation is onsiere in the Arbitrary-Lagrangian-Eulerian (ALE) framework with the onstraint that the mesh an the flui veloity oinie. The link between the vertex veloity an the flui motion is obtaine by a formulation of the momentum onservation on a lass of multi-sale enase volumes aroun mesh verties. The vertex veloity is erive with a noal Riemann solver onstrute in suh a way that the mesh motion an the fae fluxes are ompatible. Finally, the resulting sheme onserves both momentum an total energy an, it satisfies a semi-isrete entropy inequality. The numerial results obtaine for some lassial 2D an 3D hyroynami test ases show the robustness an the auray of the propose algorithm. Introution The main feature of the Lagrangian numerial methos lies in the fat that the motion of the flui is intrinsially linke to the geometrial transformation that follows flui path. This ensures that there is no mass flux rossing the bounary of ontrol volume moving with the flui. Thus, interfaes in multiimensional flows are sharply resolve. In this framework, one has to isretize not only the gas ynamis onservation laws but also the points motion in orer to move the mesh. At the isrete level, the most natural framework employs staggere-gri hyroynamis (SGH). The term staggere refers to spatial entering in whih position, veloity an kineti energy are entere at points, while ensity, pressure an internal energy are within ells. The one-imensional sheme was evelope by the pioneer work of von Neumann an Rihtmyer [32] an its two-imensional extension was ahieve by Wilkins [33]. In its original version this finite ifferene sheme has some rawbak relate to onservation loss, ontrol of entropy proution via artifiial visosity an spurious gri istortion. Many improvements have been mae in orer to inrease the auray an the robustness of SGH isretization. In [2], artifiial gris istortion an hourglass-types motions are uner ontrol by means of Lagrangian sub-zonal masses an pressures. The onstrution of a ompatible SGH isretization [0] leas to a sheme that onserves total energy. Moreover, this ompatible erivation allows for the speifiation of fores suh as those erive from an artifiial visosity or sub-zonal pressures. Finally, the isretization of artifiial visosity has been onsierably improve: first by introuing formulations for

3 multiimensional shok wave omputations [9] an then by eriving a tensorial artifiial visosity with a mimeti finite ifferene isretization [7]. The three-imensional extension of the SGH isretization an be foun in [] an also in [6]. In [28, 27], the variational multi-sale stabilize (VMS) approah was applie in finite element omputations of Lagrangian hyroynamis. In that ase, a pieewise linear ontinuous approximation was aopte for the variables. The ase of Q/P0 finite element is investigate in [29] wherein the kinemati variables are represente using a pieewise linear ontinuous approximation while the thermoynami variables utilize a pieewise onstant one. To apture shok wave, VMS metho nees a tensorial artifiial visosity. It shows promising results for two an three-imensional shok omputations. However, it must be note that it annot properly preserve interfae in the ase of multimaterial flows. An alternative to the previous isretizations is to use a ell-entere isretization in whih all the physial variables (ensity, momentum, pressure, total an internal energy) are within the ell. The fluxes an the noe isplaement are both ompute using Riemann problems at interfaes. This metho for Lagrangian gas ynamis in one imension, has been first introue by Gounov, see [7] an [26]. Its two-imensional extension has been performe uring the eighties, [2] an [6]. This sheme is a ell-entere finite volume sheme on moving struture or unstruture meshes. It is onstrute by integrating iretly the system of onservation laws on eah moving ell. The flux aross the bounary of the ell is ompute by solving exatly or approximately a one-imensional Riemann problem in the iretion normal to the bounary. The main problem with the two-imensional version lies in the fat that the noe veloity neee to move the mesh annot be iretly alulate. In [2], the noe veloity is ompute via a speial least squares proeure. It onsists in minimizing the error between the normal veloity oming from the Riemann solver an the normal projetion of the vertex veloity. It turns out that it leas to an artifiial gri motion, whih requires a very expensive treatment [5]. Moreover, with this approah the flux alulation is not onsistent with the noe motion. Reently, new ell-entere methos have been propose in [9] an []. These new approahes use a fully Lagrangian form of the gas ynamis equations, that is, the graient an ivergene operators are expresse in the Lagrangian oorinates. This type of isretization nees to follow the Jaobian matrix assoiate to the map between Lagrangian an Eulerian spaes. In [9], the oeffiients of the Jaobian matrix are introue as new inepenent variables loate within the ells. The vertex veloity is obtaine by inverting ompatibility onitions. In this ase, the noe isplaement is not onsistent with the flux omputation. To solve this problem Després an Mazeran in [3] use the free ivergene onstraint to isretize the oeffiients of the Jaobian matrix at the interfaes. They show that it amounts to ompute the noe veloity in a onsistent way. The key point in this work is the introution of noal Riemann solver suh as to erive the noal veloity oherently with the interfae fluxes. The global onservation of momentum an total energy is ahieve an semi-isrete entropy inequality is provie. However, this noal solver exhibits a strong epenene to the ell aspet ratio whih an lea to severe numerial instabilities. This iffiulty is ritial for Lagrangian hyroynamis omputations an thus has motivate Maire et al. [22] to propose an alternative sheme that suessfully solves the aspet ratio problem an in the same time keeps the onsisteny between the noe isplaement an the fluxes omputation. The main new feature of this algorithm is the introution of four pressures on eah ege, two for eah noe on eah sie of the ege, this is the main ifferene from [3]. This sheme also loally onserves momentum, total energy an it satisfies a loal entropy inequality. In the present paper, the Lagrangian formulation is onsiere in the Arbitrary-Lagrangian- Eulerian (ALE) framework with the onstraint that the mesh an the flui veloity oinie. We evelop a finite volume formulation for unstruture gris, wherein ontrol volumes are mae of polygons in 2D an polyhera in 3D. The only assumption neee for our sheme is that the faes of the ells must be ( ) simpliies, where is the imension of the spae. 2

4 For = 2 this assumption is always satisfie, for = 3 we split eah fae of the polyheral ells into triangles. The link between the vertex veloity an the flui motion is obtaine by a formulation of the momentum onservation on a lass of sub-sales enase volumes aroun mesh verties. This lea to a balane relation between sub-zonal fores whih are ompute by half-riemann problems to efine verties motion an ontrol the numerial visosity. In orer to ensure onservation properties an onsisteny with the Geometrial Conservation Law (GCL), veloity an pressure are interpolate on interfaes with two ifferent sets of linear funtions, satisfying an orthogonality property. The resulting numerial fluxes are ompatible with the mesh motion. Moreover, in two imensional ase, this formulation reovers the shemes propose in [3] an [22]. The paper is organize as follows. First, we erive the spatial first orer isretization of the gas ynamis equations in Lagrangian form an give its main features. Then, we show how to onstrut the seon orer extension thanks to pieewise linear monotoni reonstrution. We ahieve the erivation of the numerial sheme by eveloping its seon orer time isretization an by giving some pratial issues relate to the time step limitations. Last, we valiate our new sheme with 2D an 3D test ases. They are representative test ases for ompressible flui flows an emonstrate the robustness an the auray of this new sheme. 2 Spatial isretization 2. Governing equations Let D be an open subset of IR, = 2 or 3, fille with an invisi ieal flui an equippe with an orthonormal frame. We are intereste in isretizing the equations of the Lagrangian hyroynamis. It is onvenient, from the point of view of subsequent isretization to write the unsteay ompressible Euler equations in the ontrol volume formulation whih hols for an arbitrary moving ontrol volume t t t V (t) V (t) V (t) ρ V + S(t) ρu V + ρe V + S(t) S(t) ρ(u κ) N S = 0, ρu(u κ) N S = ρe(u κ) N S = S(t) S(t) PN S, PU N S. Here, V (t) is the moving ontrol volume, an S(t) its bounary. ρ, U, P, E are the mass ensity, veloity, pressure an speifi total energy of the flui. N enotes the unit outwar normal vetor to the moving bounary S(t) whose veloity is enote by κ (kinemati veloity). Impliit in the use of these equations is also the onservation of volume: t V (t) V V (t) (a) (b) () κ N S = 0. (2) This equation is also name geometri onservation law (GCL) an, it is equivalent to the loal kinemati equations X = κ, X(0) = x, (3) t where X are oorinates efining the ontrol volume surfae at time t > 0 an x are the oorinates at time t = 0. Then, X = X(x,t) are impliitly efine by the loal kinemati equations, whih are also name the trajetory equations. This enables us to efine the map M t : V (0) V (t) x X(x,t) 3

5 where X is the unique solution of (3). With fixe t, this map avanes eah flui partile from its position at time t = 0 to its position at time t. Let us introue F = x X, the Jaobian matrix of this map an J its eterminant. Then, time ifferentiation of J gives the lassial equation J J κ = 0, t whih is nothing but the loal version of the GCL equation (2). The thermoynamial losure of the set of equations () is obtaine by the aition of an equation of state whih is taken to be of the form P = P(ρ,ε), (4) where the speifi internal energy, ε, is relate to the speifi total energy by ε = E 2 U 2. The set of previous equations is referre to as the Arbitrary Lagrangian Eulerian (ALE) integral form of the Euler equations an an be foun in many papers [2, 8]. Equations (a), (b) an () express the onservation of mass, momentum an total energy. In the Lagrangian formalism the rates of hange of volume, mass, momentum an energy are ompute assuming that the omputational volumes are following the material motion. This leas to the following set of equations t t t V (t) V (t) V (t) ρ V = 0, ρu V + ρe V + S(t) S(t) PN S = 0, Pκ N S = 0, where the kinemati veloity κ is obtaine from the kinemati onstraint (5a) (5b) (5) X S(t), κ(x,t) = U(X,t). (6) We notie that equation (5a) implies that the mass of the ontrol volume remains onstant. 2.2 Notations an assumptions Let us onsier the physial omain V (0) that is initially fille by the flui. We assume that we an map it by a set of ells without gaps or overlaps. Eah ell may be a general polygon in 2D or a general polyheron in 3D. It is assigne a unique inex, an is enote by Ω (0). For a given time t > 0, we set Ω (t) = M t [Ω (0)], where M t is the map previously efine. Here, we assume that Ω (t) is still a polygon in 2D or a polyheron in 3D, that is, the map M t is a ontinuous an linear funtion over eah element of the mesh. Here, we have use the term polyheron to esribe the types of three-imensional ells over whih we isretize the onservation laws. Our efinition of a polyheral ell is a volume enlose by an arbitrary number of faes, eah etermine by an arbitrary number (3 or more) of verties. If a fae has four or more verties, they an be non-oplanar, thus the fae is not a plane an it is not possible to efine its unit outwar normal. In orer to avoi this problem, we eie to ivie eah fae into triangular faets without aing supplementary verties beause we o not want to a supplementary unknowns. For instane, in the ase of an hexaheron ell, eah quarangular fae is ivie into two triangular faes. For the subsequent isretization, we assume that the bounary, Ω (t), of the ell is the reunion of faes whih are ( )-simplies, that is, segment for = 2 an triangles for = 3. Eah vertex of the mesh is assigne a unique inex p an we enote by C(p) the set of ells that share a partiular vertex p. We subivie eah ell into a set of sub-ells. Eah sub-ell is 4

6 Ω p p Figure : Fragment of a 2D unstruture gri, inluing ell Ω an point p. The soli lines efines the primal gri, an the ashe lines show the meian mesh. The meian mesh is forme by onneting the ell enters,, to the mi-sie points,. The ashe area shows the sub-ell Ω p. uniquely efine by a pair of inies an p an is enote by Ω p = Ω p. In 2D, this sub-ell is onstrute by onneting the entroi of the ell Ω to the mipoints of ell eges impinging on point p, see Figure. In orer to efine the sub-ell in 3D we must efine the following auxiliary points: the entroi of the ell, the entroi of eah fae an the mipoint of eah ege impinging on point p. In aition, these points are onnete by straight lines. The reunion of the sub-ells Ω p that share a partiular point p enables to efine the vertex-entere ell, Ω p, relate to the vertex p Ω p = Ω p. (7) C(p) With the previous notations, we have introue a primal mesh forme by the ells Ω an a ual one forme by the vertex-entere ells Ω p. This ual mesh is also name meian mesh. Here, we use ompletely similar notations to those introue in [8]. We enote by Ω p an Ω p the bounaries of the ells Ω p an Ω p. In orer to improve the reaability of the paper we give hereafter the list of the ifferent sets of inex that will be use throughout the paper. Let us note that the generi inies utilize in the sequel are for the ells, p for the points (verties) an f for the faes. Sets relate to the ell : P() enotes the set of points of the ell ; F() enotes the set of faes of the ell ; F p () enotes the set of faes of the ell that share point p. Sets relate to the point p: C(p) enotes the set of ells that share point p; F(p) enotes the set of faes that share point p. Sets relate to the fae f: P(f) enotes the set of points of the fae f. 5

7 Sets relate to the sub-ell p: F (p) enotes the set of faes of the sub-ell p that share the entroi of ell ; F p (p) enotes the set of faes of the sub-ell p that share the point p. Using the previous notations it is straightforwar to show that F() = F p (p). (8) 2.3 Disretization of the GCL p P() In orer to efine a numerial sheme for the GCL we have to properly efine an approximate map M t. For sake of simpliity we first make the supplementary assumption that the ells are simpliies. Therefore, a ontinuous an linear map over eah element will preserve the mesh struture. Using the finite element formalism, this transformation is efine on the ell Ω (t) by the position of its verties X p (t) X (x,t) = ϕ p (X)X p (t), (9) p P() where P() is the set of verties of ell Ω (t) an ϕ p (X) is the baryentri oorinate relate to vertex p. Sine the transformation is linear, the baryentri oorinate is invariant, that is, ϕ p (X) = ϕ p (x). Therefore, the approximate kinemati veloity fiel is obtaine by time ifferentiation of (9) κ (X,t) = ϕ p (x)κ p (t), (0) p P() where κ p is the veloity of vertex p. The Jaobian matrix F of this transformation is written F = κ p (t) x ϕ p, p P() where x is the graient operator efine with the x oorinates. F is a onstant funtion sine ϕ p is linear. Therefore, its eterminant J is also a onstant funtion over the ell Ω, whih is equal to the volume of the ell. The GCL equation is written V t = κ N S, Ω (t) = κ N S. S f Here, V = Ω f F() V enotes the volume of the ell Ω, κ is efine by (0) an F() enotes the set of faes of Ω. We notie that eah fae S f of the ell Ω is a ( ) simplex an, the restrition of κ over the fae S f is linear. Therefore, the integral in the previous equation is written κ N S = S f ( κ p ) S f N f, p P(f) where P(f) is the set of verties of fae S f an N f its unit outwar normal. Combining the previous results we obtain the isretization of the GCL equation for a simpliial mesh V t f F() ( κ p ) S f N f = 0. () p P(f) 6

8 This funamental equation esribes the Lagrangian representation of flui flow. It is generalize to the ase of a non simpliial mesh in 2D (resp. in 3D) by triangularizing (resp. tetraheralizing) polygons (resp. polyhera) of arbitrary orer so that the volume variation of any ells an be ompute in a general manner on an unstruture gri. Comment. The GCL isretization an be utilize in a ifferent way by notiing that equation () etermines the isrete form of the ivergene of the veloity fiel over the ell Ω ( κ) = V V t = V f F() = V p P() ( κ p ) S f N f p P(f) ( S f N f) κ p. f F p() Here, F p () enotes the subset of faes of Ω that share point p. By efining the orner vetor, Γ p, that is assoiate with ell an point p we finally obtain Γ p = f F p() ( κ) = V p C(p) S f N f, Γ p κ p. (2) This last equation has been previously erive an utilize in [0, ] in orer to onstrut ompatible hyroynamis algorithms using the metho of Support Operators. 2.4 Disretization of the physial onservation laws The aim of this setion is to provie a isretization for the system of the physial onservation laws. Setting q = 0 U, φ(q,n) = PN, (3) E Pκ N the system (5) an be written in the general form ρq V + t Ω (t) Ω (t) φ(q,n) S = 0. (4) Here, we have use the ontrol volume efine by the primal ell Ω. We notie that the first equation of the above system orrespons to mass onservation. Its integration provies ρ V = m, Ω (t) where m is the onstant mass of the ell Ω. Let us efine the mass average value of q over the ell Ω q = ρq V. m Ω By applying this efinition for q = ρ we get the following form of mass onservation V (t) = m ρ (t), (5) 7

9 where ρ is the mean value of the mass ensity over ell Ω. By using (5), we notie that the GCL an be rewritten m t ( ) κ N S = 0. (6) ρ Ω (t) Therefore, the GCL an be viewe also as a physial onservation law relate to the speifi volume ρ. With the previous omments the system of physial onservation laws has the generi form m t q + φ(q,n) S = 0, (7) where we have set q = ρ U E Ω (t), φ(q,n) = κ N PN Pκ N (8) We assume that the bounary, Ω (t), of ell is the reunion of faes S f (t) whih are ( )- simplies. Thus, the physial onservation laws beome where the fae flux φ f is efine by m t q + φ f = S f (t) f F() φ f = 0, (9) φ(q,n) S. Sine the fae S f (t) is a ( )-simplex its unit outwar normal is onstant, i.e. N(X,t) = N(t). Therefore, the numerial flux is written φ f = κ f S fn f P f S fn f (Pκ) f S fn f. (20) κ f, P f an (Pκ) f enote the fae fluxes efine by κ f = S f (t) Pf = S f (t) (Pκ) f = S f (t) S f (t) S f (t) κ(x, t) S, P(X,t) S, S f (t) (2a) (2b) (P κ)(x, t) S. (2) Finally, we obtain the following set of isrete equations for the isrete variables ( ρ,u,e ) m t ( ρ ) f F() m t (U ) + f F() m t (E ) + f F() S f N f κ f = 0, S f N fp f = 0, S f N f (Pκ) f = 0. (22a) (22b) (22) 8

10 The points motion is given by the isrete trajetory equation t X p = κ p, X p (0) = x p, (23) where κ p κ p ( ρ,u,e ) is the point veloity. We notie that all primary variables, inluing material veloity, are ell-entere as it is one in [2, 22]. In orer to omplete this isretization the following important problems arise: How o we ompute the fae fluxes efine by system (2)? How o we ompute the point veloities κ p? These veloities being known, how an we ensure the ompatibility between the mesh motion an the volume variations of the ells? We have alreay answere to the last question. In fat we have shown previously that (22a) is fully equivalent to the GCL provie the fae flux κ f is written κ f = p P(f) κ p. (24) This funamental relation enables to write two equivalent isretizations of the speifi volume variation. It an be given in term of the flux through the faes, or equivalently, in term of vertex fluxes. Moreover, the two isretizations are ompatible with the point motion uner the onition (24). Hene, one an onsier two methos for omputing the fae veloities: The first one relies on the evaluation of the normal veloity using a one-imensional Riemann solver at faes. The vertex veloities have to be ompute by solving a linear system built from equations (24) written for all the faes. This system is in general singular, that is why we give up this approah an shall aopt a more robust metho. In the seon metho, the point veloities are first evaluate using some still-to-efine solver. The fae veloities are then ompute from (24). This is the tehnique we will use in the sequel of the paper beause this metho will guarantee the ompatibility between vertex motion an ell volumes variation. Thus, our task onsists in builing a numerial solver that an ompute the fae fluxes P f, (Pκ) f an the point veloity κ p. We resolve these questions in the next setions. 2.5 Derivation of the point veloity We efine a strategy to ompute the noal veloity. We reall that the kinemati veloity is assume to be ontinuous an linear in orer to efine a oherent map M t that preserves the struture of the mesh. One natural way to obtain the noal veloity onsists in hoosing the ual ell Ω p instea of the primal ell Ω as a ontrol volume for the momentum equation t ρκ V + Ω p PN S = 0. Ω p (25) It leas to a staggere spatial plaement of the variables wherein the position an the veloity are efine at gri points, an ensity, internal energy, an the pressure are efine at ell enters [0]. Here, we will proee ifferently by keeping a entere spatial plaement of the physial variables (ensity, momentum, pressure, total energy) an in the same time efining oherently 9

11 κ p P p Ω p Ω q Figure 2: Initial onitions for the multi-imensional Riemann problem at point p. the kinemati veloity to move the mesh. Consier a point p an the ells C(p) that share this point. In eah ell the flui flow is haraterize by the onstant state q = ( ρ,u,e ) t. One way to ompute the point veloity κ p onsists in solving the multi-imensional Riemann problem at point p efine by the initial onitions q, C(p), see Figure 2. Its solution woul provie also the point pressure P p, whih is the instantaneous value of pressure at point p immeiately following the breakown of the initial isontinuity. Knowing κ p an P p, we oul easily ompute the fae fluxes φ f an esign a onservative sheme. Unfortunately, suh an approah is not possible sine up to our knowlege the solution of suh a multi-imensional Riemann problem is still not known The lassial approah Dukowiz an its o-authors [2] suggest an alternative approah base on one-imensional Riemann problems. Let us onsier an 2, two ajaent ells that share point p, see Figure 3. We enote by f the ommon fae to these ells an by N f, N2 f the unit outwar normals relate to this fae. We set N f = N 2 f = N f. Here, the one-imensional Riemann problem is efine by the isontinuity of the state variables q on either sie of the ell fae f, in the viinity of point p. The solution of this Riemann problem provies the unique pressure Ppf an normal veloity κ pf N f of the ontat surfae. In the ase of an approximate aousti Riemann solver, these values satisfy the following linear system [4] P P pf = Z (κ pf U ) N f, P 2 P pf = Z 2 (κ pf U 2 ) N f, (26a) (26b) where Z i for i =,2 is the aousti impeane. A straightforwar alulation shows that P pf = Z P 2 + Z 2 P Z + Z 2 Z Z 2 Z + Z 2 (U U 2 ) N f, κ pf N f = (Z U + Z 2 U 2 ) N f Z + Z 2 P P 2 Z + Z 2. Knowing P pf an κ pf N f one gets immeiately the fae fluxes however, the point veloity is still unknown. In [2], κ p is taken to be the vetor whose omponents normal to ajaent ell 0

12 Ω 2 q 2 N f κ p N 2 f Ω q Figure 3: Initial onitions for the one-imensional Riemann problem at fae f in the viinity of point p. faes agrees with the Riemann veloity at eah ajaent ell fae, in a weighte least squares sense κ p = argmin ω f (κ p N f κ pf N f ) 2, f F(p) where F(p) is the set of faes that share point p an ω f is a positive weight efine in [2]. By notiing that in general κ p N f κ pf N f, we realize that this way of eriving the point veloity leas to an inonsisteny with the GCL isretization. In aition, with suh an approah the kinemati veloity κ will be isontinuous in the viinity of point p A new approah To ensure onsisteny with the GCL isretization an ontinuity of the kinemati veloity aroun point p, we propose to solve two half one-imensional Riemann problems at eah ell interfae by assuming that its veloity is equal to the point veloity κ p. Thus, at eah fae f onnete to point p, we introue two interfae pressures Pp an P f p 2f, see Figure 4, efine by P P p,f = Z (κ p U ) N f, P 2 P p 2f = Z 2 (κ p U 2 ) N f. (27a) (27b) By substrating (27b) from (27a), we get P p 2f P p f = (Z + Z 2 )(κ pf N f κ p N f ). We notie that Pp = P f p if an only if 2f κ pf N f = κ p N f. In this ase, we reover the solution of the one-imensional Riemann problem an we get Pp = P f p = P 2f pf an κ pf N f = κ p N f = κ pf N f. Sine in general κ pf N f κ p N f we obtain the isontinuity Pp P f p. 2f Finally, for eah simpliial fae f we introue 2 fae pressures Pp an P f p 2f for p P(f), from whih we are able to ompute the fae fluxes. The isontinuity of these pressures implies the loss of momentum onservation for our ell-entere isretization. Moreover, the point

13 f 3 f 4 q 3 κ p P p2 f 2 q 2 f 2 q 4 P p f 2 P p f q f Figure 4: Loalization of the multiple pressures for the half Riemann problems in the viinity of point p. veloity has not been yet etermine. We will show that these two ritial issues-momentum onservation an point veloity omputation- are strongly linke. Sine the momentum onservation equation, in its ALE form, (b), is vali for any moving ontrol volume V, we an write it for the ual ell, Ω p, an for all the subells, Ω p, that surroun point p t t ρu V + ρu(u κ) N S + PN S = 0, (28a) Ω p Ω p Ω p ρu V + ρu(u κ) N S + PN S = 0, C(p). (28b) Ω p Ω p Ω p We notie that (28a) expresses momentum onservation over the ual ell, whereas (28b) expresses momentum balane loally over eah subell. Knowing that Ω p = Ω p, Ω p = S p f, C(p) C(p) f F (p) where F (p) is the subset of faes of Ω p that share point, equation (28a) is rewritten ρu V + ρu(u κ) N S + PN S = 0. (29) t Ω p S f S f C(p) f F (p) The subivision of the subell bounary Ω p = S p f f F (p) f F (p) S p f f F p(p) where F p (p) is the subset of faes of Ω p that share point p, enables to rewrite (28b) as ρu V + ρu(u κ) N S + PN S t Ω p f F S p(p) f f F S p(p) f + ρu(u κ) N S + PN S = 0, C(p). S f S f f F (p) f F (p), 2

14 We notie that the seon term in the left-han sie is equal to zero sine Ω is a Lagrangian ell (U = κ for f F p (p)). The summation of the previous equation over all the subells that surroun point p yiels ρu V + ρu(u κ) N S + PN S + t C(p) Ω p f F S (p) f f F S (p) f PN S = 0. S f C(p) f F p(p) Finally, by substrating this last equation from (29) we obtain By setting the equation (30) an be rewritten C(p) f F p(p) S f PN S = 0. (30) Ppf = P(X,t) S, S f (t) S f (t) C(p) f F p(p) S f P pfn f = 0. (3) Note that this equation is invariant by any homothety entere at point p. It expresses the balane of momentum loally aroun point p. Hene Ppf an be viewe as a noal pressure loate at point p an relate to ell an fae f. Sine the veloity of fae f in the viinity of point p is equal to the noal veloity κ p, the noal pressure Ppf is ompute using the following half approximate Riemann problem P P pf = Z (κ p U ) N f, for f F p (p). (32) Here, Z enotes the aousti impeane efine in ell an N f is the unit outwar normal relate to the fae S f for f F p (p). The substitution of (32) into (3) leas to S f Z (N f N f)κ p = [ Sf P N f + S f Z (N f N ] f)u. (33) C(p) f F p(p) C(p) f F p(p) The noal veloity satisfies a system. Sine the geometri variables (surfae S f an unit outwar normal N f) epen on the noal veloity κ p via the trajetory equation (23), we notie that the previous system is non linear. This noal solver is the multi-imensional extension of the two-imensional solver erive in [22] Summary By setting M pf = S f Z (N f N f), for f F p (p), an M p = C(p) f F p(p) M pf, (34) the point veloity κ p an the point pressure Ppf relate to ell an fae f are written κ p = M [ p Sf P N ] f + M pf U, C(p) f F p(p) P pf = P Z (κ p U ) N f, for f F p (p). We note that the matrix M p is symmetri positive efinite an thus always invertible. 3

15 Comment 2. Here, we have erive the noal veloity utilizing the aousti approximate Riemann solver for whih Z = ρ a, where a is the loal spee of soun. As suggeste by Dukowiz [4] we an use the artifiial shok visosity approximation by rewritting (32) as P P pf = Z pf (κ p U ) N f, for f F p (p), (35) [ with Z pf = ρ a + A (κ p U ) N f ], where A is a material-epenent parameter that is given in terms of the ensity ratio in the limit of very strong shoks. In the ase of a gamma law gas one gets A = γ+ 2 where γ is the polytropi inex. Comment 3. Instea of using the unit outwar normal, N f, relate to the fae S f in equation (32), one an introue the average orner normal N p efine by S p N p = S f N f. f F p(p) Then, there is only one half Riemann problem orresponing to this unit normal an, it is written P P p = Z (κ p U ) N p. (36) This amounts to efine only one noal pressure Pp for eah ell that surrouns point p. With this hoie, the system satisfie by the noal veloity is written S p Z (N p N p )κ p = [S p P N p + S p Z (N p N p )U ]. (37) C(p) C(p) Using the orner normal N p in the efinition of the half Riemann problem (32), we have reovere the noal solver evelope in [3]. The fore orresponing to the single noal pressure P p reas F p = S p P pn p. We note that this subell fore is always olinear to the geometri iretion N p. Due to this fat, it appears that the noal solver propose in [3] exhibits a strong epenene to the ell aspet ratio. This problem an lea to numerial instability. 2.6 Fluxes approximation The aim of this setion is to provie an approximation of the fae fluxes κ f, P f, (Pκ) f for the isrete equations (22) relate to the physial onservation laws. The fae fluxes approximations are onstrute by using a linear mapping over eah simpliial faes. This linear mapping utilizes the noal veloity an pressures provie by the noal solver. We shall show also that this approximation enables us to reover momentum an total energy onservation. Let us onsier a ell an one fae f F(). Sine S f is a ( )-simplex, for eah p P(f) we introue, ϕ p, the baryentri oorinate relate to the vertex p. We reall the lassial result ϕ p (X)S = S f S f. (38) 2.6. GCL flux In orer to be self-onsistent, we reall briefly the fae flux approximation that orrespons to the GCL equation. We have shown previously that the veloity over the fae S f is written κ(x,t) = ϕ p (X)κ p (t), for X S f p P(f) The substitution of the veloity fiel into the efinition of the fae flux (2a) leas to κ f = κ p. (39) p P(f) 4

16 2.6.2 Momentum flux The omputations of the momentum an total energy fluxes are mae using the following linear interpolation for the pressure P(X,t) = ψ p (X)Ppf(t), for X S f. p P(f) Here, ψ p is a linear funtion over S f to be etermine. Sine the noal pressure Ppf is isontinuous aross S f we enote the fae flux Pf instea of P f. After substitution of the linear interpolation we get Pf = P(X,t)S S f S f = α p Ppf, p P(f) where the unknown oeffiient α p is written α p = ψ p (X)S. S f S f We laim that with this approximation of the momentum flux, momentum onservation is ensure. Omitting the bounary onitions an summing the momentum equation on eah ell, written with the previous approximation of the momentum flux, we get the global balane of momentum t ( m U ) = S f N fα p Ppf f F() p P(f) = S f N fα p Ppf thanks to (8) p P() f F p(p) = α p S f N fppf p = 0. C(p) f F p(p) Here, we have replae the global summation over ells by a global summation over points an, we have use the fat that Ppf satisfies equation (3) Total energy flux We ompute the total energy flux interpolation by ombining the linear interpolations of the veloity an the pressure (Pκ)(X,t) = ψ q (X)ϕ p (X)Pqf(t)κ p (t), for X S f. p P(f) q P(f) Sine the noal pressure Ppf is isontinuous aross S f we enote the total energy fae flux (Pκ) f instea of (Pκ) f. After substitution of the linear interpolation we get (Pκ) f = (Pκ)(X,t)S S f S f = β pq Pqfκ p, p P(f) q P(f) 5

17 where the unknown oeffiient β pq is written β pq = ϕ p (X)ψ q (X)S. S f S f We give a suffiient onition to ensure total energy onservation. We laim that if the oeffiients β pq are written uner the form β pq = C p δ p,q, (40) where C p is an unknown oeffiient an δ p,q the Kroneker symbol, then the total energy onservation is ensure. The emonstration onsists in writing the global balane of total energy t ( m E ) = = = = p = 0. f F() p P(f) q P(f) f F() p P(f) p P() f F p(p) C p C(p) f F p(p) S f N fβ pq P qf κ p S f N fc p P pf κ p thanks to (40) S f N fc p P pf κ p thanks to (8) S f N fppf κ p We note that the term between the parenthesis is null beause of the momentum onservation Definition of the ψ p funtions Finally, it remains to etermine the linear funtion ψ p for p P(f) in orer to ompute the oeffiients α p an C p. This etermination is performe using the following eomposition of the unknown funtion over the basis {ϕ p }for p P(f) ψ p = q P(f) A pq ϕ q, p P(f), (4) where A pq are the 2 unknown oorinates of ψ q funtion. These oorinates are ompute using the two following onitions Consisteny onition: {ψ p }for p P(f) must preserve onstant funtions, that is, p P(f) ψ p =. The substitution of ψ p in terms of its oorinates gives the equivalent onition p P(f) Energy onservation onition: ψ q must satisfy the onition ϕ p (X)ψ q (X)S = C p δ p,q. S f S f A pq =, q P(f). (42) 6

18 By substituting ψ q in terms of its oorinates an using the result ϕ p (X)ϕ r (X)S = + δ p,r S f S f ( + ), we obtain S f S f ϕ p (X)ψ q (X)S = r P(f) A qr ( + δ p,r ) ( + ). Hene, we fin that the oorinates must satisfy A qr ( + δ p,r ) = 0, (p,q) P 2 (f)an p q. (43) r P(f) The ombination of (42) an (43) gives us a linear system of 2 equations for the 2 unknown oorinates A pq. We an show that the eterminant of this system is always non null, thus this system amits a unique solution whih is written { if p = q, A pq = if p q. Therefore the funtions ψ p are written ψ p = ( + )ϕ p. (44) We notie that the basis {ψ p } an be viewe as the ual basis of {ϕ p }. Finally, knowing the ψ p funtions, we get α p = an β pq = δp,q Summary With the previous linear interpolations the fae fluxes are written κ f = κ p, P f = p P(f) (Pκ) f = p P(f) P pf, p P(f) P pfκ p. (45a) (45b) (45) We notie that this formulation of the fae fluxes leas to a numerial sheme that onserves momentum an total energy. 2.7 The semi-isrete evolution equations We give in this setion the summary of the semi-isrete evolution equations that onstitute a lose set of equations for the unknowns ( ρ,u,e ): m t ( ρ ) m t (U ) + m t (E ) + f F() p P(f) f F() p P(f) f F() p P(f) S f N f κ p = 0, S f N fp pf = 0, S f N f P pfκ p = 0. (46a) (46b) (46) 7

19 We reall that the point veloity κ p an the point pressure Ppf are written κ p = M p C(p) f F p(p) [ Sf P N f + M pf U ], (47a) P pf = P Z (κ p U ) N f, for f F p (p), (47b) where the matries M pf an M p are efine by (34). The motion of the mesh is rule by the semi-isrete trajetory equation t X p = κ p, X p (0) = x p. Comment 4. In the Lagrangian formalism we have to onsier two types of bounary onitions on the borer of the omain D: either the pressure is presribe or the normal omponent of the veloity. Here, we o not etail the implementation of these bounary onitions. Let us notie that they are onsistent with our noal solver. For a etaile presentation about this topi the reaer an refer to [22]. 2.8 Entropy inequality for the semi-isrete sheme We show that our sheme in its semi-isrete form satisfies a loal entropy inequality. We ompute the time variation of the speifi entropy σ in ell using the Gibbs formula σ m T t = m [ ε t + P t ( )], (48) ρ where T enotes the mean temperature of the ell. Thanks to the efinition of the internal energy this equation is rewritten σ m T t = m [ E U U + P t t t ( )]. ρ We ot-multiply equation (46b) an substrat it from the total energy equation (46) an we get m [ E U U t t ] = S f N fppf (κ p U ). f F() p P(f) The pressure work is ompute by multiplying (46b) by P an it is written P t ( ρ ) = = f F() p P(f) f F() p P(f) S f N fp κ p S f N fp (κ p U ). The last line of the previous equation omes from the fat that for a lose polyheron we have S f N f = 0. f F() Finally, the ombination of the previous results yiels m T σ t = f F() p P(f) S f N f(p P pf) (κ p U ). (49) 8

20 Now, using the efinition of the pressure flux (47b) we get m T σ t = f F() p P(f) S f Z [(κ p U ) N f] 2. This equation represents a loal entropy inequality for the semi-isrete sheme sine its righthan sie is always positive. Comment 5. The entropy proution of our semi-isrete entere sheme has a struture very similar to the artifiial visosity term use in staggere sheme [9]. But, we must amit that our entropy proution term is always ative even in the ase of isentropi flows. For suh flows our sheme oes not onserve entropy. This property is typial from Gounov-type shemes. However, this extra entropy proution an be ramatially erease by using a seon orer extension of the sheme. 3 Spatial seon orer extension The spatial seon orer extension is obtaine by a pieewise linear monotoni reonstrution of the pressure an veloity, given by their mean values over mesh ells [4, 3]. 3. Pieewise linear reonstrution Let u u(x) enotes a flui variable (pressure or veloity omponents), we assume a linear variation for u in ell u (X) = u + u (X X ). (50) Here, u is the mean value of u in ell an u is the graient of u that we are looking for. We note that X = V Ω X V is the ell entroi so that the reonstrution is onservative. The graient in (50) is ompute by imposing that u (X ) = u for C(), where C() is the set of neighboring ells of ell. This problem is generally overetermine an thus the graient is obtain by using a least squares proeure. Hene, it is the solution of the following minimization problem u = argmin [u u u (X X )] 2. C() A straightforwar omputation shows that this solution is written u = M (u u )(X X ), (5) where M is the matrix given by M = C() C() (X X ) (X X ), we notie that M is symmetri positive efinite an thus always invertible. The main feature of this least squares proeure is that it is vali for any type of unstruture mesh an moreover it preserves the linear fiels. 9

21 3.2 Monotoniity To preserve monotoniity, we limit the value that the graient is allowe to take, using the Barth Jespersen multiimensional extension [4] of the van Leer s lassial metho. For eah ell, we introue the slope limiter φ [0,] an the limite reonstrute fiel u lim (X) = u + φ u (X X ), (52) where u enotes the approximate graient given by (5). The oeffiient φ is etermine by enforing the following loal monotoniity riterion u min u lim (X) u max, X. (53) Here, we have set u min = min(min C(),u ) an u max = max(max C(),u ). Sine the reonstrute fiel is linear we note that it is suffiient to enfore the following onitions at the points u min u lim (X p ) u max, p P(). (54) so that the quantity u in the ell oes not lie outsie the range of the average quantities in the neighboring ells. Thanks to this formula we an efine the slope limiter as knowing that φ = min p P() φ,p µ( umax u u (X p) u ) if u (X p ) u > 0, φ,p = µ( umin u u (X p) u ) if u (X p ) u < 0, if u (X p ) u = 0. Here, µ enotes a real funtion that haraterizes the limiter. By setting µ(x) = min(, x) we reover the Barth Jespersen limiter. We an also efine a smoother -in the sense that it is more ifferentiable- limiter by setting µ(x) = x2 +2x x 2 +x+2. This limiter has been introue by Vankatakrishnan [3] in orer to improve the onvergene towars steay solutions for the Euler equations. Finally, instea of using the mean values of the pressure an the veloity in our noal solver, we use their noal extrapolate values eue from the linear monotoni reonstrution. 4 Time isretization For the time isretization of the semi-isrete evolution equations (46a)-(46) we use a seon orer Runge-Kutta sheme. We assume to know the physial properties in the ell an its geometrial harateristis at the beginning of the time step t n, i.e., ρ n,u n,e n,p n an X n p, for p P(). We want to ompute their values at time t n+ an we introue the time step t = t n+ t n. We esribe hereafter the preitor an the orretor steps of the seon orer Runge-Kutta time isretization. 4. Preitor step We start with the following preitor step. We enote with the supersript n +, the values at the en of this preitor step. We ompute the noal values with the noal solver: knowing the physial variables an the geometry at time t n we ompute the noal veloity κ n p by solving the linear system (33), then we eue the noal pressure relate to fae f an ell, P,n pf, thanks to the equation (32). 20

22 point motion: we avane the points position using the trajetory equation. X n+, p = X n p + tκ n p We upate the geometry an ensity: knowing X n+, p partiularly the volume of the ell V n+, mass onservation. We upate the momentum an the total energy by solving m (U n+, m (E n+, U n ) + t E n ) + t we ompute the geometry an an eue from it the new ensity ρ n+, from f F() p P(f) f F() p P(f) We upate the speifi internal energy an the pressure Sf n N,n f P,n pf = 0, S n f N,n f P,n pf κn p = Corretor step ε n+, P n+, = E n+, 2 Un+, 2, = P(ρ n+,,ε n+, ). We use the preit values in orer to omplete the time isretization We ompute the noal values with the noal solver: knowing the physial variables an the geometry at the en of the preitor step we ompute the noal veloity κp n+, by solving the linear system (33), then we eue the noal pressure relate to fae f an ell, P,n+, pf, thanks to the equation (32). point motion: we avane the points position using the trajetory equation with the entere veloity. X n+ p = X n p + t 2 (κn p + κ n+, p ) We upate the geometry an ensity: knowing X n+ p we ompute the geometry an partiularly the volume of the ell V n+ an eue from it the new ensity ρ n+ from mass onservation. We upate the momentum an the total energy by using the entere fluxes m (U n+ m (E n+ U n ) + t 2 E n ) + t 2 f F() p P(f) f F() p P(f) (S n f N,n f P,n pf + Sn+, f (Sf n N,n P,n pf κn p + S n+, f We upate the speifi internal energy an the pressure f ε n+ = E n+ 2 Un+ 2, P n+ = P(ρ n+,ε n+ ). N,n+, f P,n+, pf ) = 0, N,n+, f P,n+, pf κp n+, ) = 0. 2

23 Comment 6. We note that we oul also use the time isretization of the GCL (46a) to ompute the upate value of the ensity. However, sine the geometrial part of the volume variation flux is a quarati funtion of time, one nees to perform the time integration exatly in orer to ensure the onsisteny between the isretize GCL an the ell volume variation [23, 24]. Comment 7. The geometrial part of the momentum an total energy fluxes is isretize in an expliit manner in orer to preserve the ompatibility with the noal solver. Thus, the sheme onserves exatly momentum an total energy. 4.3 Time step limitation The time step is evaluate following two riteria. The first one is a stanar CFL riterion whih guaranties heuristially the monotone behavior of the entropy. The seon is more intuitive, but reveals very useful in pratie: we limit the variation of the volume of ells over one time step CFL riterion We propose a CFL like riterion in orer to ensure a positive entropy proution in ell uring the time step. At time t n, for eah ell we enote by λ n the minimal value of the istane between two points of the ell. We efine t E = C E min where C E is a stritly positive oeffiient an a is the soun spee in the ell. The oeffiient C E is ompute heuristially an we provie no rigorous analysis whih allows suh formula. However, extensive numerial experiments show that C E = 0.25 is a value whih provies stable numerial results. We have also heke that this value is ompatible with a monotone behavior of entropy. The rigorous erivation of this riterion oul be obtaine by omputing the time step whih ensures a positive entropy proution in ell from time t n to t n Criterion on the variation of volume λ n a n, We estimate the volume of the ell at t = t n+ with the Taylor expansion V n+ = V n + t V (t n ) t. Here, the time erivative t V is ompute by using (). Let C V be a stritly positive oeffiient, C V ]0,[. We look for t suh that V n+ V n V n C V. To o so, we efine V n t V = C V min t V (t n ). For numerial appliations, we hoose C V = 0.. Last, the estimation of the next time step t n+ is given by t n+ = min ( t E, t V,C M t n ), (55) where t n is the urrent time step an C M is a multipliative oeffiient whih allows the time step to inrease. We generally set C M =.0. 22

24 6 analytial seon orer 5 4 ρ e e e e r Figure 5: 2D Seov problem on a Cartesian gri: ensity map (left) an ensity in all the ells (right) at t =. 5 Numerial results In this setion, we present several test ases in orer to valiate our numerial sheme. For eah problems, we use a perfet gas equation of state whih is taken to be of the form P = (γ )ρε, where γ is the polytropi inex. 5. 2D Seov problem This test ase esribes the evolution of a blast wave in a point symmetri explosion for a gas haraterize by γ = 7 5. An exat solution with ylinrial symmetry is erive with self-similarity arguments in [20]. The initial ensity has a uniform unit istribution, an, the pressure is 0 6 everywhere, exept in the ell ontaining the origin. For this ell we efine P = (γ )ρ E0 V where E 0 = is the total amount of release energy. For this value, it is shown in [20] that the exat solution is a ylinrially symmetri iverging shok whose front is at raius r = (x 2 + y 2 ) = an has a peak ensity of 6. First, we perform a omputation on the [0,.2] [0,.2] quarant, subive into squares. The results obtaine in Figure 5 are quite goo an they assess the ability of the metho to respet the ylinrial symmetry. We notie that these results are very lose to those obtaine in [2] for the same setup but using a staggere sheme. For the same problem, we have also isplaye in Figure 6 the results obtaine on a polygonal gri whih is efine in [2]. One more the result shows a goo agreement with the analytial solution. The numerial metho preserves very well the one-imensional ylinrial solution. This emonstrates the ability of our solver to hanle unstruture mesh D Noh problem This test ase has been introue by Noh in [25]. A gas with γ = 5 3 is given an initial unit inwar veloity. A irular shok wave is generate whih at time t = 0.6 has a raius of 0.2. The initial thermoynami state is given by (ρ,p) = (,0 6 ). The initial omain is efine by [R,θ] [0;] [0, Π 2 ] where the polar oorinates are given by r = x 2 + y 2 an θ = artan( y x ). 23

25 6 analytial seon orer 5 4 ρ e-003.8e e e r Figure 6: 2D Seov problem on a polygonal gri: ensity map (left) an ensity in all the ells (right) at t =. We use an non-onformal mesh with two levels of refinement. This non-onformal gri ontains a mixture of triangles, quarangles an pentagons as it an be seen in Figure 7. In Figure 7, we observe the goo quality of the mesh after shok refletion an the goo agreement with the analytial solution for the ensity profile. These numerial results show the ability of our Lagrangian sheme to hanle non-onformal gri without any speifi moifiations D Saltzmann problem We onsier now the motion of a planar shok wave on a Cartesian gri that has been skewe. This test ase has been initially efine for two-imensional flows in [5]. This is a well known iffiult test ase that enables to valiate the robustness of a Lagrangian sheme when the mesh is not aligne with the flui flow. Here, we onsier the three-imensional extension of this test that has been propose in []. The omputational omain is the volume efine by (x,y,z) [0,] [0,0.] [0,0.]. The initial mesh is obtaine by transforming a uniform Cartesian gri with the mapping x sk = x + (0. z)( 20y)sin(xπ), for 0 y 0.05 x sk = x + z(20y )sin(xπ), for 0.05 y 0. y sk = y, z sk = z. We notie that this skewe gri is base on generalizing the two-imensional Saltzmann gri in the following manner: the y = 0 surfae is the original two-imensional skewe gri, this gri is aitionally skewe with respet to y oorinate. We note that the y = 0.05 surfae is not skewe at all. For this problem we use the polytropi inex (γ = 5/3). The initial state is (ρ 0,P 0,V 0 ) = (,0 6,0). At the plane x = 0, a unit normal veloity is speifie. On all the other bounaries, we set refletive bounary onitions. The exat solution is a planar shok wave that moves at spee 4/3 in the x iretion. The shok wave hits the fae x = at time t = 0.75, the ensity shoul be equal to 4 in the shoke region. We perform two omputations with our three-imensional sheme. The first one is one by using a 3D gri whih is 2D skewe, that is, we set only one ell in the y iretion an rewrite 24

26 8 6 analytial numerial R Figure 7: 2D Noh problem on a non-onformal gri: gri (left) an ensity in all the ells (right) at t = analytial 3D result with 2D skewe gri analytial 3D result with 3D skewe gri x x Figure 8: 3D Saltzmann problem: ensity in all the ells at time t = 0.7 for 2D (left) an 3D (right) eformation. x sk = x+(0. z)sin(xπ). This problem is a sanity hek that orrespons to the 2D Saltzamnn problem. The gri at time t = 0.7 is isplaye in Figure 9 (left). We notie that our sheme preserves very well the one-imensional solution. Moreover, as it an be seen in Figure 8 (left), the loation of the shok wave an shok plateau are in goo agreement with the analytial solution. These results emonstrate that our 3D sheme behaves similarly with the 2D sheme erive in [22]. The seon omputation is performe by using the 3D skewe gri efine previously. The gri at time t = 0.7 is shown in Figure 9 (right). The gri over whih the shok has propagate is less skewe than the original one. However, we note that some istortions have ourre in the viinity of the shok front. These istortions will lea to a breakown of the omputation after the first boune at time t = After this time the omputation stops beause the time step beomes too small. This too small time step ours when two points are too lose to eah other. The plot of the ensity profile in Figure 8 (right) assesses that the one-imensional solution is quite well preserve. The shok level is not uniform but it osillates slightly aroun the exat value. 25

27 Figure 9: 3D Saltzmann problem: mesh at time t = 0.7 for 2D (left) an 3D (right) eformation D Seov problem As it is sai in [30], the Seov test in three imensions is muh more hallenging than its twoimensional ounterpart, ue to the inrease istortion unergone by the elements. Here, we are going to ompute the Seov blast test on the [0,.2] [0,.2] [0,.2] otant. We use two gris mae of an ubes. The initial onitions are the same than in the 2D Seov test. The total amount of energy release in the ell ontaining the origin is ompute so that the shok wave reahes the raius r = at time t =. Following [20] we set E 0 = We suessfully run to ompletion omputations for both gris with suh an initialization. The ensity in all the ells as a funtion of the raius of the enter of the ell is isplaye in Figure 0. We note a very goo agreement with the analytial solution. Moreover, for the oarser gri our result seems to reah the same level of auray than the one obtaine in [30] with almost the same gri. For the finer gri we observe the onvergene of the numerial solution towar the analytial one. The gri an the ensity ontour are isplaye in perspetive view in Figure for the oarse gri an in Figure 2 for the fine gri. 26

28 6 6 analytial 20*20*20 gri analytial 40*40*40 gri r r Figure 0: 3D Seov problem: ensity in all the ells at time t = as a funtion of the raius of the ell enter, mesh (left) an mesh (right). Figure : 3D Seov problem: ensity ontour (left) an mesh (right) at time t = for the mesh 27

29 Figure 2: 3D Seov problem: ensity ontour (left) an mesh (right) at time t = for the mesh 28