Abstract
We present **a** **high**-**order** **cell**-**centered** **Lagrangian** **scheme** **for** solving the **two**-**dimensional** gas dy- namics equations on **unstructured** **meshes**. **A** node-based discretization of the numerical fluxes **for** the physical conservation laws allows to derive **a** **scheme** that is compatible with the geomet- ric conservation law (GCL). Fluxes are computed using **a** nodal solver which can be viewed as **a** **two**-**dimensional** extension of an approximate Riemann solver. The first-**order** **scheme** is conserva- tive **for** momentum and total energy, and satisfies **a** local entropy inequality in its semi-discrete form. The **two**-**dimensional** **high**-**order** extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in **order** to as- sess this new **scheme**. The results obtained **for** various representative configurations of one and **two**-**dimensional** **compressible** **fluid** **flows** show the robustness and the accuracy of our new **scheme**. Key words: **Lagrangian** hydrodynamics, **cell**-**centered** **scheme**, Generalized Riemann problem, **compressible** flow, **high**-**order** finite volume methods, **unstructured** mesh

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UMR CELIA, Universit´ e Bordeaux I, 351 Cours de la Lib´ eration, 33 405 Talence, France
Abstract
The goal of this paper is to present **high**-**order** **cell**-**centered** schemes **for** solving the equations of **Lagrangian** gas dynamics written in cylindrical geometry. **A** node-based discretization of the nu- merical fluxes is obtained through the computation of the time rate of change of the **cell** volume. It allows to derive finite volume numerical schemes that are compatible with the geometric con- servation law (GCL). **Two** discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume **scheme** while the second one corresponds to the so-called area weighted **scheme**. Both formulations share the same discretization **for** the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as **a** **two**-**dimensional** extension of an approximate Rie- mann solver. The control volume **scheme** is conservative **for** momentum, total energy and satisfies **a** local entropy inequality in its first-**order** semi-discrete form. However, it does not preserve spher- ical symmetry. On the other hand, the area weighted **scheme** is conservative **for** total energy and preserves spherical symmetry **for** one-**dimensional** spherical flow on equiangular polar grid. The **two**-**dimensional** **high**-**order** extensions of these **two** schemes are constructed employing the general- ized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in **order** to assess these new schemes. The results obtained **for** various representative configurations of one and **two**-**dimensional** **compressible** **fluid** **flows** show the robustness and the accuracy of our new schemes.

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Key words : Finite volume methods, **unstructured** grids, anisotropic diffusion, parallel computing.
1 Introduction
In this paper, we describe **a** finite volume **scheme** to solve anisotropic diffusion equations on **unstructured** grids. This three-**dimensional** **scheme** is the natural extension of the **two**- **dimensional** **scheme** CCLAD (**Cell**-**Centered** **LAgrangian** Diffusion) initially presented in [27]. We aim at developing **a** robust and flexible method **for** diffusion operators devoted to the numerical modeling of the coupling between heat transfers and **fluid** **flows**. More precisely, we are concerned by the numerical simulation of heat transfers in the domain of hypersonic ablation of thermal protection systems [11]. In this context, one has to solve not only the com- pressible Navier-Stokes equations **for** the **fluid** flow but also the anisotropic heat equation **for** the solid materials which compose the thermal protection. These **two** models, i.e., the Navier- Stokes equations and the heat equation, are strongly coupled by means of **a** surface ablation model which describes the removal of surface materials resulting from complex thermochem- ical reactions such as sublimation. We point out that in our case, the Navier-Stokes equations

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traction boundary conditions **for** all the domain boundaries except the left one which is enforced to be **a** wall boundary. We recall this problem has neither an analytical solution nor experimental results. However, it is **a** good test to assess the robustness of our numerical method, while keeping in mind that the numerical solution accuracy will be limited by equations model studied, which is in our case the **compressible** gas dynamics system. In Figure 15, we have displayed the initial grid and the final density maps obtained at time t = 0.005 by means of the first-**order** and limited second- **order** DG schemes, with the acoustic solver, on **a** 100 × 10 Cartesian grid. Comparing Figures 15(b) and 15(c), we clearly see the second-**order** final solution grid **a** lot more deformed than the first-**order** one. This phenomenon can be explained by the large amount of numerical diffusion inherent of the first-**order** **scheme**. As it has been demonstrated in [43] in the case of **a** Gresho-like vortex problem, in some extreme cases the first-**order** **scheme** is unable to simulate appropriately the problems to the final time due to the large numerical diffusion. But obviously, resolving the **compressible** gas dynamics equations, the numerical schemes presented are not able to capture elastic waves and features. This is the reason why the second-**order** results, Figure 15(c), are quite far from the ones obtained in [31] with an elastic-plastic second-**order** **cell**-**centered** **Lagrangian** **scheme**. Nevertheless, this test case permitted us once more to prove the robustness of the **cell**-**centered** **Lagrangian** schemes presented, as no negative density or negative internal energy appears during the calculation. 13. Conclusion

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of neighboring cells sharing **a** node, one cannot apply in **a** straightforward manner one-**dimensional** solvers to define uniquely the grid point velocity. The staggered hydrodynamics has been developed to avoid such complications. In this framework, **a** staggered discretization is employed such that the kinematic variables (vertex position, velocity) are located at the nodes whereas the thermody- namic variables (density, pressure, internal energy) are defined at the **cell** centers. The conversion of kinetic energy into internal energy through shock waves, consistent with the second law of thermo- dynamics, is ensured by adding an artificial viscosity term. The staggered grid schemes employed in most hydro-codes have been remarkably successful over the past decades in solving complex multi-**dimensional** **compressible** **fluid** **flows**, refer **for** instance to [45, 88, 17, 18, 32, 55, 33, 34, 4]. However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetry breaking. In addition, the fact that all variables are not conserved over the same space can make these schemes difficult to handle when one wants to assess analytical properties of the numerical solution. **For** all these reasons, this paper focuses on the **cell**-**centered** approach. Different techniques may be employed to build the numerical fluxes and move the grid through the use of approximate Riemann solvers, with respect to the GCL. The interested readers may refer to the following papers [1, 19, 20, 61, 72, 15, 53, 82, 5, 16, 10, 84] **for** **a** more detailed description of this approach and its variants. Let us mention that besides these **two** mainly employed approaches, i.e. the **cell**-**centered** and staggered approaches, **a** third one has recently grows quickly in popular- ity these past years. This third framework, referred to as Point-**Centered** Hydrodynamic (PCH), combines the features of the first **two**, namely **a** dual grid and the fact that kinetical and thermody- namical variables are conserved on the same cells. Indeed, in this particular frame, the momentum and total energy conservation equations are solved on the dual grid around the nodes, generally by means of an edge-based finite element **scheme** or an edge-based upwind finite volume method. The PCH approach has been successful applied these past decades to problematics concerned with the simulation of incompressible **flows**, **compressible** **Lagrangian** **flows**, or **Lagrangian** solid dynamics, refer **for** instance to [28, 29, 24, 42, 48, 76, 79, 78, 86, 87, 67, 68, 2, 3]. **Two** of the main advantages of these schemes are that there are very well adapted to triangular or tetrahedral grids, as well as they reduce in most cases problems related to mesh stiffness.

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one-**dimensional** solvers to define uniquely the grid point velocity. The staggered hydrodynamics has been developed to avoid such complications. In this framework, **a** staggered discretization is employed such that the kinematic variables (vertex position, velocity) are located at the nodes whereas the thermodynamic variables (density, pressure, internal energy) are defined at the **cell** centers. The conversion of kinetic energy into internal energy through shock waves, consistent with the second law of thermodynamics, is ensured by adding an artificial viscosity term. The staggered grid schemes employed in most hydro-codes have been remarkably successful over the past decades in solving complex multi-**dimensional** **compressible** **fluid** **flows**, refer **for** instance to [33, 73, 10, 11, 45]. However, they clearly have some theoretical and practical deficiencies such as mesh imprinting and symmetry breaking. In addition, the fact that all variables are not conserved over the same space can make these schemes difficult to handle when one wants to assess analytical properties of the numerical solution. **For** all these reasons, this paper focuses on the **cell**-**centered** approach. Different techniques may be employed to build the numerical fluxes and move the grid through the use of approximate Riemann solvers, with respect to the GCL. The interested readers may refer to the following papers [1, 12, 13, 52, 63, 8, 41, 70, 3, 9, 7, 72] **for** **a** more detailed de- scription of this approach and its variants. In the section dedicated to the **two**-**dimensional** case, **a** general procedure to develop first-**order** finite volume schemes on general polygonal **meshes** is presented. Such formulation will cover the numerical methods introduced in [12, 55, 8, 70, 72]. Although **a** wide range of purely **Lagrangian** formulations are available, together with the different advantages associated to them, it is well known these descriptions admit **a** severe drawback in some situations. In the presence of intense vortexes or

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This paper aims at presenting **a** **cell**-**centered** indirect ALE algorithm to solve multi-material **compressible** **flows** on **two**-**dimensional** **unstructured** grids with fixed topology. Our **Lagrangian** phase solves the gaz dynamics equations utilizing **a** moving mesh **cell**-**centered** discretization wherein the physical conservation laws are discretized in **a** compatible manner with the nodal velocity so that the geometric conservation law (GCL) is exactly satisfied [12]. Namely, the time rate of change of **a** **Lagrangian** volume is computed consistently with the node motion. This critical requirement is the cornerstone of any **Lagrangian** multidimensional **scheme**. Nowadays, **cell**-**centered** finite volume schemes [11, 36, 35] that fulfill this GCL requirement seem to be **a** promising alternative to the usual staggered finite difference discretization [10]. Moreover, these **cell**-**centered** schemes allow straightforward implementation of conservative remapping methods when they are used in the context of ALE. Here, we are using the **high**-**order** **cell**-**centered** **Lagrangian** **scheme** that has been described in [35]. Let us recall that the numerical fluxes are determined by means of **a** node-**centered** approximate Riemann solver. This discretization leads to **a** conservative and entropy consistent **scheme** whose **high**-**order** extension is derived through the use of generalized Riemann problem [7, 35].

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is an accurate and robust method, which can produce impressive results, even on **unstructured** polygonal grids, see **for** instance [21].
An alternative to the staggered discretization is to use **a** conservative **cell**-**centered** discretization. This method **for** **Lagrangian** gas dynamics in one dimension, has been introduced by Godunov, see [12] and [24]. The multidimensional extension of this method has been performed during the eighties, [2] and [11]. This multidimensional **scheme** is **a** **cell**-**centered** finite volume **scheme** on moving structured or **unstructured** **meshes**. It is constructed by integrating directly the system of conservation laws on each moving **cell**. The primary variables, density, momentum and total energy are defined in the cells. The flux across the boundary of the **cell** is computed by solving exactly or approximately **a** one-**dimensional** Riemann problem in the direction normal to the boundary. The main problem with this type of method lies in the fact that the node velocity needed to move the mesh cannot be directly calculated. In [2], the node velocity is computed via **a** special least squares procedure. It consists in minimizing the error between the normal velocity coming from the Riemann solver and the normal projection of the vertex velocity. It turns out that it leads to an artificial grid motion, which requires **a** very expensive treatment [10]. Moreover, with this approach the flux calculation is not consistent with the node motion.

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sin θ,
where (r, θ) are the polar coordinates corresponding to the Cartesian coordinates (x, y). The corresponding distorted grid obtained setting a0 = 0.25 and n = 8 is plotted in Figure 22(b). Now, we compute the numerical solution of the non-linear test problem on the above distorted grid, using **two** different schemes. These are: the classical five-point **scheme** and the CCLADS **scheme** which reduces to **a** nine-point **scheme** on quadrangular grids. The numerical solution resulting from the five-point **scheme** is plotted in Figure 23 using blue dots. We have displayed the temperatures in all cells as function of the **cell** center radius versus the reference solution. In this figure, we observe the main flaw of the five-point **scheme**: in spite of its robustness, it produces **a** numerical solution wherein the temperature front is aligned with the grid distortion. The corresponding numerical solution is not able to preserve the cylindrical symmetry. In addition, the comparison to the reference solution, shows that the timing of the thermal wave is completely wrong. Let us emphasize that this test case is not **a** fake problem. It is representative of situations which frequently occur in the framework of plasma physics simulation wherein the heat conduction equation is coupled with **a** numerical method solving **Lagrangian** hydrodynamics equations. In this case, grid distortions are induced by **fluid** motion and thus the use of the five-point **scheme** to solve the heat conduction equation leads to **a** very bad result. This weakness of the five-point **scheme** follows from the fact that its construction is based on **a** **two**-point flux approximation 3 , which becomes inaccurate in the

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non structur´ es bidimensionnels
R´ esum´ e : In this paper, we describe **a** **cell**-**centered** **Lagrangian** **scheme** devoted to the numeri- cal simulation of solid dynamics on **two**-**dimensional** **unstructured** grids in planar geometry. This numerical method, utilizes the classical elastic-perfectly plastic material model initially proposed by Wilkins [M.L. Wilkins, Calculation of elastic-plastic flow, Meth. Comput. Phys. (1964)]. In this model, the Cauchy stress tensor is decomposed into the sum of its deviatoric part and the thermodynamic pressure which is defined by means of an equation of state. Regarding the devi- atoric stress, its time evolution is governed by **a** classical constitutive law **for** isotropic material. The plasticity model employs the von Mises yield criterion and is implemented by means of the radial return algorithm. The numerical **scheme** relies on **a** finite volume **cell**-**centered** method wherein numerical fluxes are expressed in terms of sub-**cell** force. The generic form of the sub-**cell** force is obtained by requiring the **scheme** to satisfy **a** semi-discrete dissipation inequality. Sub-**cell** force and nodal velocity to move the grid are computed consistently with **cell** volume variation by means of **a** node-**centered** solver, which results from total energy conservation. The nominally second-**order** extension is achieved by developing **a** **two**-**dimensional** extension in the **Lagrangian** framework of the Generalized Riemann Problem methodology, introduced by Ben-Artzi and Fal- covitz [M. Ben-Artzi and J. Falcovitz, Generalized Riemann Problems in Computational **Fluid** Dynamics, Cambridge Monographs on Applied and Computational Mathematics, 2003]. Finally, the robustness and the accuracy of the numerical **scheme** are assessed through the computation of several test cases.

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Figure 7 depicts the error distribution in the plane z = 0.009 of the display cube, **for** the conservative transport of **a** SQUARE. In the ISE S panel, there is no conservativity-correction applied at each time iteration, while in the ISE Z panel, the conservativity-correction procedure is activated. We clearly see that the error is much bigger **for** the DON **scheme**. To be convinced of this fact, let us proceed to 1-D cuts along the 3 principal directions of the display cube. In Fig. 8, we compare the 3 schemes in each cut panel. As far as the amplitude is concerned, ISE S and ISE Z are quite similar. As **for** DON, it is obviously worse by one **order** of magnitude. Figures 9 and 10 plot the same results but associated with the WAVELET data. The observa- tions are even more impressive here, insofar as no **scheme** is good enough to predict to maximal value, but DON remains the most diffusive. One can wonder why the situation is worse **for** WAVELET, **a** smooth function, than **for** SQUARE, **a** discontinuous data. The answer is: although we are working with highly accurate schemes, we are still very far from convergence, because the (deformed) mesh we are using is very coarse with respect to the data.

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100 kpc-scale **high**-density arms to exist in the cluster environment
(Sanders et al. 2013). The strength and topology of magnetic fields are responsible **for** determining the propagation of cosmic rays in galaxies (Strong & Moskalenko 1998). Magnetic fields are also present in **a** wide variety of stars and play **a** significant role in their evolution (Donati & Landstreet 2009). Many of these problems are suitable **for** study by **a** moving mesh approach and up until now **a** divergence-free MHD solver has been lacking. This paper lays the framework **for** implementing **a** divergence-preserving algorithm **for** robust evolution of the MHD equations in moving mesh codes such as AREPO (Springel 2010) and TESS (Duffell & MacFadyen 2011). In Section 2 we describe the details of the numerical method. In Section 3 we show the results of numerical tests (with compar- isons to fixed grid CT and the Powell cleaning **scheme** on **a** moving mesh) demonstrating that the method works well and has several advantages over the other **two** techniques. In Section 4 we discuss variations of our method and future directions. In Section 5 we pro- vide concluding remarks.

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on **a** staggered mesh. The staggered location of the velocity unknowns allows to compute **a** free-divergence velocity field, up to the computer accuracy, which is used as transport velocity in the mass conservation equation. Numerical simulations of Rayleigh-Taylor instabilities are performed and presented. The first **order** rate of convergence with respect to the time step is recovered both in the case of Newtonian and Bingham **flows**. Comparisons with published results in the case of Newtonian **flows** validate the parallel implementation of the bi-projection **scheme**. Numerical simulations of **a** Bingham flow at Bi = 10 and 100 are reported and discussed. From the best of our knowledge, these results are the first ones obtained **for** this problem. At even larger Bingham number, we may expect that the heavier **fluid** will not flow. Investigating in more details the dynamics and behavior of Rayleigh-Taylor instabilities when the Bingham number is increased could be the scope of further works.

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Diffuse interface methods with **compressible** fluids, considered through hyperbolic multiphase flow models, have demonstrated their capability to solve **a** wide range of complex flow situations in severe conditions (both **high** and low speeds). These formulations can deal with the pres- ence of shock waves, chemical and physical transformations, such as cavitation and detonation. Compared to existing approaches able to consider **compressible** materials and interfaces, these methods are conservative with respect to mixture mass, momentum, energy and are entropy preserving. Thanks to these properties they are very robust. However, in many situations, typ- ically in low transient conditions, numerical diffusion at material interfaces is excessive. Several approaches have been developed to lower this weakness. In the present contribution, **a** specific flux limiter is proposed and inserted into conventional MUSCL type schemes, in the frame of the diffuse interface formulation of Saurel et al. (2009). With this limiter, interfaces are captured with almost **two** mesh points at any time, showing significant improvement in interface represen- tation. The method works on both structured and **unstructured** **meshes** and its implementation in existing codes is simple. Computational examples showing method capabilities and accuracy are presented.

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velocity’s unknowns are associated with the vertices of the cells, while the ther- modynamics’s unknowns (internal energy, density) are associated with the cells of the mesh. This is **a** natural choice **for** semi-**Lagrangian** methods, because it directly provides **a** procedure to move the mesh compatibly with the velocity unknowns. This intrinsic quality justifies the great number of works devoted to this method **for** more than 60 years. Among these works, some concern the design of artificial viscosity. Artificial viscosity is needed in staggered schemes, because the discretization of the pressure gradient is by construction "**centered**" and does not account **for** the entropy increase in shock waves. The artificial viscosity cures this flaw in adding **a** dissipative term proportional to the ve- locity jump into the momentum equation. Modern versions of this mechanism include **a** monotonic second-**order** reconstruction of this jump, in **order** to dis- card the artificial viscosity **for** regular (isentropic) **flows**. One indirect and never mentioned consequence of this enhancement is to widely improve the behavior of staggered schemes in low-Mach regimes. Indeed, it has been shown that **a** correct calculation of low-Mach **flows** is directly related to **a** **centered** pressure gradient (refer **for** instance to [ 19 ]).

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the maximum pressure and impact duration differed from one identical wave impact to the next, even in carefully controlled laboratory experiments, while the pressure impulse appears to be more repeatable. **For** sure, the simple one-**fluid** models which are commonly used **for** examining the peak impacts are no longer appropriate in the presence of air. There are few studies dealing with **two**-**fluid** models. An exception is the work by Peregine and his collaborators. Wood, Peregrine & Bruce [4] used the pres- sure impulse approach to model **a** trapped air pocket. Peregrine & Thais [5] examined the effect of entrained air on **a** particu- lar kind of violent water wave impact by considering **a** filling flow. Bullock et al. [6] found pressure reductions when compar- ing wave impact between fresh and salt water where, due to the different properties of the bubbles in the **two** fluids, the aeration levels are much higher in salt water than in fresh. H. Bredmose recently performed numerical experiments on **a** **two**-**fluid** system which is quite similar to the one we will use below. He developed **a** finite volume solver **for** aerated **flows** named Flair [7].

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Unite´ de recherche INRIA Lorraine, Technoˆpole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, IRI[r]

The nature of the 𝜁 **scheme** and, in particular, the version of this **scheme** called 𝜁 simplified **scheme** by Laurent and Nguyen [1], makes it an ideal candidate **for** its extension to **unstructured** grids with cells of arbitrary shapes because the **scheme** relies on **a** traditional MUSCL reconstruction, and only requires **a** local additional limitation to be applied to the reconstructed quantities on **cell** faces. The extension of this **scheme** to **unstructured** grids and its implementation into the OpenFOAM framework [56] are the topics of this article, the remainder of which is organized as follows: in Sec. 2 the problem of moment transport is introduced. The 𝜁 simplified **scheme** **for** hexahedral structured grids of Laurent and Nguyen [1] is summarized in Sec. 3. Its generalization to **unstructured** grids with cells of arbitrary shapes is discussed in Sec. 4. Finally, the same one- and **two**-**dimensional** test cases used by Laurent and Nguyen [1] are used in Sec. 5 to verify the implementation of the numerical **scheme**, using hexahedral uniform grids. Then, the **two**-**dimensional** cases are repeated using **a** triangular grid, and the **order** of accuracy of the 𝜁 simplified **scheme** on **unstructured** grids is assessed.

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The model is based on an artificial linearized pressure law **for** low Mach regime. This artificial pressure law induces several interesting effects from mathematical and numerical viewpoints described above and in previous related papers. The novelty of the present work is to extend the multi-scale adaptive **scheme** proposed by [10] to our three-**dimensional** hyperbolic equations. The automatic mesh refinement process is performed thanks to an original criterion: the numerical density of entropy production. The increase of computational costs caused by the re-meshing procedure is outweighed by the introduction of an efficient Block-Based Adaptive Mesh Refinement method (BB-AMR) allowing an easy and fast parallel implementa- tion and computation. Through several 2D and 3D wave impact test cases, the overall numerical approach is validated. In particular, the entropy production appears to be **a** relevant criterion **for** automatic mesh refinement process, which leads to better describe the regions of interest (interfaces and impact zones) and to drastically reduce computational time.

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along the boundary . Ex
ept at the forefront stagnation point, the entropy de-
viation of the third **order** s
heme is mu
h
loser than the exa
t one.
We have re-run this test
ase on an hybrid mesh using the se
ond **order** and
the third **order** s
hemes. In both
ases, the same degrees of freedom are used