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Keywords: second-order finite volume; **unstructured** **meshes**; anisotropic diffusion; ALE framework; fluid dynam- ics; coupling.
1 INTRODUCTION
In this talk we describe two cell-centered finite volume **schemes** developed in an Arbitrary Lagrangian Eulerian (ALE) framework for solving respectively anisotropic diffusion equation and Navier-Stokes equations. We then describe an ablation model that is used as a boundary condition in the two **schemes** to introduce the coupling between the different phenomenons.

2. Both P1-AP and diffusion **schemes** may adapt as well to cartesian **meshes** as to **unstructured** **meshes** with straight or conical edges (note also that ω can be non homogeneous and then a local ω can be used on each edge).
3. We note the unnatural property: increasing the geometric order allows for a stabilization of a lower order scheme. A similar phenomenon was observed in [5, 6, 4] for non-linear or transport advection equations. In our case, we observe it both in explicit and implicit version of the **schemes**.

Upwind Schemes for the Two-Dimensional Shallow Water Equations with Variable Depth Using Unstructured Meshes Alfredo Bermúdez, Alain Dervieux, Jean-Antoine Desideri, Maria Elena Vázquez.[r]

1 Number of degrees of freedom for third and fourth order approximation for triangular **meshes**. 59
2 Number of verti
es, triangles and quadrangles for the dierent **meshes** used for the grid
onvergen
e. The left number in the
olumn T riangles
orresponds to the number of triangles in the triangular mesh, while the right one is the number of triangles in the hybrid grid. Hybrid grids have then about two times less elements than the triangular twin ones. 59
3 L
2

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2.3.2. The step scheme
Much less sophisticated **schemes** are also available that are worth a detailed attention in our specific context. Con- sidering our constraints in terms of computational times, the use of the exponential scheme will indeed be only justified if it insures significantly higher accuracy levels for com- bustion applications. In 2000 [15], J. Liu used the “step” scheme, which corresponds to the “Upwind” scheme that is commonly used in CFD. This scheme had already been proposed by Chai et al., in 1995 [9], in order to solve the RTE by the FVM for irregular geometries using curvilin- ear coordinates. In previous studies, this scheme has often been applied to Cartesian structured grids, in order to avoid negative values that could occur with **schemes** such as the “diamond” one [16]. Omitting the scattering phenomenon, the intensity I P is evaluated at the center of the cell by ap-

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to 1%.
Conclusion, extensions
In this work, an admissibility and AP scheme for system of conservation laws of type ( 1.1 ) on any **unstructured** mesh is developed, using the DLP scheme [ 21 ] as a limit scheme. The introduction of the source term is done using the tech- nique developed in [ 7 ], and the preservation of the set of admissible states A is guaranteed thanks to an a posteriori correction in the spirit of the MOOD method [ 18 ]. This a posteriori correction uses a physical admissibility detector that can be changed to an entropic criterion such as in [ 3 ]. The development of a high-order scheme based on the HLL-DLP scheme will be possible for instance using the MOOD method and SSP Runge-Kutta **schemes** [ 28 ] (or the **schemes**

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Abstract
In this work, we focus on the numerical approximation of the shallow water equations in two space dimensions. Our aim is to propose a well-balanced, all-regime and positive scheme. By well-balanced, it is meant that the scheme is able to preserve the so-called lake at rest smooth equilibrium solutions. By all-regime, we mean that the scheme is able to deal with all flow regimes, including the low-Froude regime which is known to be challenging when using usual Godunov-type finite volume **schemes**. At last, the scheme should be positive which means that the water height stays positive for all time. Our approach is based on a Lagrange- projection decomposition which allows to naturally decouple the acoustic and transport terms. Numerical experiments on **unstructured** **meshes** illustrate the good behaviour of the scheme.

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expected, the first-order standard scheme fails to predict the correct solution,
and even on the finer ∆x = 2.5 × 10 −3 grid, it remains worst than the first-
order bmodified scheme. Second-order standard scheme performs better, but more than respectively two or three grid refinements are necessary to achieve the same precision than respectively the emodified or the bmodified **schemes**. We also observe that the entropy enforcement for the modified scheme gener- ates a loss of accuracy corresponding to roughly one grid refinement comparing emodified with bmodified. However, the convergence rates of the different

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equations on a moving mesh such that conservation of mass, mo- mentum, and energy, and, importantly, the solenoidal (divergence- free) nature of the magnetic field is preserved. On static Carte- sian **meshes**, the constrained transport (CT) algorithm achieves this goal. It uses a finite volume formalism to evolve the density, momentum, and energy, and exploits Stokes’ theorem and uses a face-averaged representation of the magnetic fields (called the ‘staggered-mesh’ approach) to enforce ∇·B = 0 (Evans & Hawley 1988). The CT algorithm has been described in the literature as be- ing quite difficult (if not impossible) to extend to an **unstructured** mesh (Duffell & MacFadyen 2011; Pakmor, Bauer & Springel 2011; Pakmor & Springel 2013) and leading to the development of alternate divergence-cleaning **schemes** to keep ∇·B small but non-

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and the particle phase are inter-penetrating continua [6] . Its un- derlying assumption is the existence of a separation of scales: the size of the averaging region is much larger than the particle scale. This class of methods is computationally effective but the estab- lishment of an accurate continuous description of the solid phase is challenging and its formulation requires semi-empirical closures and detailed validations. On the other hand, the Discrete Element Method (DEM) also referred to as discrete particle method allows for a more detailed description of particle-particle and particle- wall interactions. This deterministic approach ﬁnds its origins in the molecular dynamics methods initiated by Alder and Wain- wright [7] and has been beneﬁting from its advances ever since. In CFD/DEM, or Euler-Lagrange methods, the gas phase is still con- sidered continuous and its time evolution is obtained from a clas- sical CFD-type Eulerian code, but the particles are described indi- vidually assuming that their motion obeys Newton’s second law of motion, which is solved using standard **schemes** for ordinary differential equations. This level of modeling designated as meso- scale still requires closures for drag, collision and other forces as a CFD grid cell typically contains up to a few tens of particles, but its advantages lie in its ability to account for the particle-wall and particle-particle interactions in a more realistic manner than Euler- Euler methods.

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The first issue concerns the lack of accuracy in the low Mach regime of Godunov-type **schemes**. While these methods performs well at capturing shocks, they may generate spurious numerical diffusion when they are used for simulating low Mach flows over relatively coarse mesh, i.e. mesh size much bigger the Mach number. Improvements of Godunov-type **schemes** more generally of collocated methods have been proposed by many authors like [29, 16, 21, 5, 9, 7, 28, 18, 24, 23, 14, 11, 8, 19]. The analysis of these authors may rely on different arguments like the analysis of the viscosity matrix [29], an asymptotic expansion in terms of Mach number [16], a detailed study in [11] that seek for invariance properties of the numerical scheme transposing the framework of Schochet [25] to the discrete setting, and also an analysis based on the so-called Asymptotic Preserving property [20] in [19]. Nevertheless the resulting cure usually boils down to reduce the numerical diffusion in the momentum equation for low Mach number values.

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.
Since C ux = C rhs , then u n r,i = u n z,i = 0 and the gas stays at rest.
3.2 A second-order scheme using the multislope MUSCL method
In the early 70's, Van Leer [11] introduced the MUSCL technique (Monotonic Upwind **Schemes** for Conservation Laws) to get a more accurate approximation with less diusion eect while maintaining stability. Extensions to multidimen- sional situations for **unstructured** **meshes** have been proposed (see [12,18]). We present here a new extension of the MUSCL technique on triangles where we use approximations of the directional derivative of U as proposed in [1315] instead of an approximation of ∇U.

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- Interface roughness effects due to mixed cells.
These issues become pregnant when dealing with **unstructured** grids as it is more difficult to control artificial smearing and roughness.
Recently a compressive limiter was introduced to sharpen diffuse interfaces in compressible two-phase flow modelling in the frame of ‘diffuse interfaces’ (Chiapolino et al., 2017, Saurel and Pantano, 2018). This limiter showed enhanced capturing properties with 2-3 cells only in the interfacial zone, when used in the frame of MUSCL type **schemes** and **unstructured** **meshes**. It is thus considered in the present contribution to solve the Level-Set function to control numerical smearing. Its ability to preserve volume and maintain shapes is examined and will be shown to be reasonably accurate.

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2. Related work
Interpolatory subdivision surface. Interpolatory subdivision surface can mainly be split into two cat- egories. In the first one, the subdivision **schemes** consistently contain the old vertices in the refined mesh after each subdivision operation. Obviously, the limit surface generated in this way will contain all the original vertices. Butterfly scheme [ Dyn et al. ( 1990 )] and Kobbelt’s scheme [ Kobbelt ( 1996 )] both fall into this category. Li et al. proposed a method for directly deducing new interpolation subdivision masks from the corresponding approximation subdivision masks [ Kobbelt ( 1996 )]. A unified interpolatory subdivision scheme is proposed for quadrilateral **meshes** based on local refinement operations in a way similar to that for approximating **schemes** generalizing splines of an arbitrarily high-order continuity in [ Deng and Ma

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The situation is much more difficult for 2D applications however. While it is quite straightforward in the case of Cartesian grids (see [5] for example), the sit- uation is way more complex on **unstructured** grids. One of the reasons is that the classical two-point flux scheme (or FV4 [23]) which is the target of many AP **schemes** is not consistent anymore. The only exception is the MPFA-based AP scheme for Friedrich systems developed in [13].

Adrien Catella, Victorita Dolean and St´ephane Lanteri
Abstract—Numerical methods for solving the time-domain Maxwell equations often rely on cartesian **meshes** and are variants of the finite difference time-domain (FDTD) method due to Yee [1]. In the recent years, there has been an increasing interest in discontinuous Galerkin time-domain (DGTD) methods dealing with **unstructured** **meshes** since the latter are particularly well adapted to the discretization of geometrical details that char- acterize applications of practical relevance. However, similarly to Yee’s finite difference time-domain method, existing DGTD methods generally rely on explicit time integration **schemes** and are therefore constrained by a stability condition that can be very restrictive on locally refined **unstructured** **meshes**. An implicit time integration scheme is a possible strategy to overcome this limitation. The present study aims at investigating such an implicit DGTD method for solving the 2D time-domain Maxwell equations on non-uniform triangular **meshes**.

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Let us note that the classical Suliciu relaxation fluxes obtained for θ = 1 do not have a truncation error that is uniform with respect to the Mach number.
6. Numerical results
We propose to test both LPS-IMEX(θ) and LPS-EX(θ) **schemes** against low Mach number test cases and order 1 Mach number test cases. LPS-EX(θ) com- putations are performed with a time step satisfying both (9) and (13), while LPS-IMEX(θ) computations are performed with a time step defined by an ex- plicit evaluation of (13) (explicit means here that u ∗ defined by (8) is used to evaluate ∆t).