In this paper, we present a conservative cell-centered Lagrangian FiniteVolume scheme for solving the hyperelasticity equations on unstructured multidimensional grids. The starting point of the present approach is the cell-centered FV discretiza- tion named EUCCLHYD and introduced in the context of Lagrangian hydrody- namics. Here, it is combined with the a posteriori Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure robustness and stability at shock waves with piecewise linear spatial reconstruction. The ADER (Arbitrary high or- der schemes using DERivatives) approach is adopted to obtain second-order of ac- curacy in time. This strategy has been successfully tested in an hydrodynamics context and the present work aims at extending it to the case of hyperelasticity. Here, the hyperelasticty equations are written in the updated Lagrangian frame- work and the dedicated Lagrangian numerical scheme is derived in terms of nodal solver, Geometrical Conservation Law (GCL) compliance, subcell forces and com- patible discretization. The Lagrangian numerical method is implemented in 3D un- der MPI parallelization framework allowing to handle genuinely large meshes. A relatively large set of numerical test cases is presented to assess the ability of the method to achieve effective second order of accuracy on smooth flows, maintaining an essentially non-oscillatory behavior and general robustness across discontinuities and ensuring at least physical admissibility of the solution where appropriate. Pure elastic neo-Hookean and non-linear materials are considered for our benchmark test problems in 2D and 3D. These test cases feature material bending, impact, com- pression, non-linear deformation and further bouncing/detaching motions.
In this paper, we describe a high-order cell-centeredfinitevolumemethod for solving anisotropic diffusion on two-dimensional unstructured grids. The resulting numerical scheme, named CCLAD (Cell-Centered LAgrangian Diffusion), is characterized by a local stencil and cell-centered un- knowns. It is devoted to the resolution of diffusion equation on distorted grids in the context of Lagrangian hydrodynamics wherein a strong coupling occurs between gas dynamics and diffusion. The space discretization relies on the introduction of two half-edge normal fluxes and two half-edge temperatures per cell interface using the partition of each cell into sub-cells. For each cell, the two half-edge normal fluxes attached to a node are expressed in terms of the half-edge temperatures impinging at this node and the cell-centered temperature. This local flux approximation can be derived through the use of either a sub-cell variational formulation or a finite difference approx- imation, leading to the two variants CCLADS and CCLADNS. The elimination of the half-edge temperatures is performed locally at each node by solving a small linear system which is obtained by enforcing the continuity condition of the normal heat flux across sub-cell interface impinging at the node. The accuracy and the robustness of the present scheme is assessed by means of various numerical test cases.
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Abstract. We present a finitevolume based cell-centeredmethod for solving diffusion equa- tions on three-dimensional unstructured grids with general tensor conduction. Our main motivation concerns the numerical simulation of the coupling between fluid flows and heat transfers. The corresponding numerical scheme is characterized by cell-centered unknowns and a local stencil. Namely, the scheme results in a global sparse diffusion matrix, which couples only the cell-centered unknowns. The space discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is charac- terized by the introduction of sub-face normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered temperature and the adjacent sub-face temperatures. The elimination of the sub-face temperatures with respect to the cell-centered temperatures is achieved lo- cally at each node by solving a small and sparse linear system. This system is obtained by enforcing the continuity condition of the normal heat flux accross each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finitevolumemethod are assessed by means of various numerical test cases. AMS subject classifications : 65M08, 65F10, 68W10, 76R50
In steady state solutions to (4.1), which by definition satisfy
∂ r F ( U ,r ) = S ( U ,r ) , (4.4)
the source terms exactly balance the flux terms. We will construct the well-balanced version of the finitevolume scheme introduced in the previous subsection by imposing that the same property must hold at the discrete level of approximation for the family of discrete steady states. For instance, cell–centered evaluation of the source terms, generally, do not ensure the preservation of these discrete steady states. Therefore, we look for an adapted discretization of the source-term which directly uses information from the steady state equation and, more specifically, uses the characterization (3.17) exhibited in Section 3, above. In turn, our scheme will satisfy a discrete version of the steady state system (4.4).
A large amount of publications are devoted to the mathematical and numerical analysis of two-phase flow models. In particular, the existence, uniqueness, and regularity of a (weak) solution are studied in [ 35 , 12 , 5 ,
6 , 14 , 15 ]. In the same spirit, much work has been carried out for developing appropriate numerical methods and proving their convergence, or a priori error estimates, like in [ 16 ] for a finite element discretization. In this paper we focus on the finitevolumemethod [ 29 ]. In this context, the convergence of a cell-centered “mathematical” scheme involving the global pressure and the Kirchhoff transform has been obtained in [ 37 ]. Alternatively, the convergence of a cell-centeredfinitevolume scheme with phase-by-phase upstream weighting (the so-called “engineering” scheme) has been shown in [ 30 ]. Vertex-centeredfinitevolume methods in the “mathematical” context have been studied in, e.g., [ 28 ], and in the “engineering” context in, e.g., [ 32 ], see also the references therein.
The numerical simulation of impacts on dissipative solids has been and is again mainly per- formed with the classical finite element method coupled with centered differences or Newmark finite difference schemes in time . Though this approach has well-known advantages, clas- sical time integrators introduce high frequency noise in the vicinity of discontinuities which is hard to remove with artificial viscosity without destroying the accuracy of the numerical so- lution. The finitevolumemethod, initially developed for the simulation of gas dynamics , has gained recently more and more interest for problems involving impacts on solid media [5, 6, 7, 8]. This method shows some advantages to achieve an accurate tracking of wavefronts; among others (i) continuity of fields is not enforced on the mesh in its cell-centered version, that allows for capturing discontinuous solutions, (ii) the characteristic structure of hyperbolic equations can be introduced within the numerical solution, either through the explicit solution of a Riemann problem at cell interfaces, or in a implicit way through the construction of the numerical scheme, (iii) the amount of numerical viscosity introduced can be controlled locally as a function of the local regularity of the solution, so that to permit the elimination of spurious numerical oscillations while preserving a high order of accuracy in more regular zones.
The scheme could still be analyzed in the general case, and its numerical properties be essentially preserved, although their proofs require much technical work, see for instance .
1.2. The discrete duality finitevolume approaches in 3D. Given a finitevolume mesh, the 2D DDFV method relies on the diamond formula  to compute gradients of the unknown u from finite differences in two independent directions, involving values of u at the centers and at the vertices of the control volumes (see [16, 10, 3]). Hence, the DDFV strategy consists in building two finitevolume meshes, namely the cell-centered given mesh and a vertex centered one. In 2D, there is a geometric duality relationship such that the interfaces between control volumes of each of these two meshes can be gathered by pairs. These pairs of edges define the so-called diamond cells, on which the gradient vectors are computed. The diamond cells are quadrilateral as shown on figure 1.1(a).
Figure 5. Bump on tail test: time evolution of the total energy for ”Banks” methods (CD4 and up5), for the unsplit Vfsl methods (Vfsl3-ns and Vfsl5-ns) and for semi-Lagrangian method (Lag3 and Lag5). N x = N v = 128, ∆t = 0.01.
see that the details have been eliminated by the scheme. When a higher order is used (as for up5 or Vfsl5- ns), additional small structures are described. On Figures 10 , 11 , we again see the link between LAG3/up3, LAG5/up5 and also CD4/PPM1 for small ∆t as depicted in Remark 3.3 . In particular, the bad oscillations of the centered reconstruction PPM1 are emphasized, when (very) small time steps are used, whereas the uncentered reconstructions LAG3 and LAG5 are insensitive to the decrease of time step. Note also that the PPM1 reconstruction behaves well when the time step is not too small, which is possible for a semi-Lagrangian scheme.
domains where the diffusion tensor S = Id. The cell-centered FV method with an
upwind discretization of the convective terms provides the stability and is extremely robust. However in this case, the mesh is assumed to be admissible [7, Definition 9.1]. In particular, this implies that the orthogonality condition has to be satisfied. As mentioned in , a difficulty in the implementation is to construct such admissi- ble meshes. Structured rectangular meshes are admissible, but they cannot be used for complex geometries arising in physical contexts. Furthermore, the finite element (FE) method allows for an easy discretization of diffusive terms with full tensors without imposing any restrictions on the meshes. However, some numerical insta- bilities may arise in the convection-dominated case.
In the present work, we concentrate on amending the vertex centeredfinite-volume formulation on triangles. A family of cell shapes is defined from a variational argument. Instead of starting from the Galerkin method for obtaining median cells, we introduce a for- mulation characterized by a modified differentiation operator, in which edgewise gradients are defined on the so-called diamond cell. The use of this diamond derivative reduces the stencil of the scheme, and in the case of cartesian triangulations, the new cell reduces to a rectangular cell. Another feature of the proposed construction is the bridge that is thus established between finitevolume, and in particular with special cell shapes as introduced by Barth , with finite element (of P1-Galerkin type) and with multidimensional schemes. We develop an analysis of the new family of schemes by proposing a necessary condition for consistency in the presence of stretched meshes. The rest of the scheme construction is more standard: an approximate Riemann solver is introduced for stability purposes, a MUSCL  interpolation is applied for increasing the formal order of accuracy, and lim- iters can be (optionally) added for saving positiveness.
Although the above-mentioned applications are quite complex, there are still some very basic problems to solve for the simulation of the Maxwell equations. This article is focused on one of them, namely the fact that fi- nite volume solutions to Maxwell equations often show some mesh dependent structures. This is due to the fact that solutions are piece-wise constant and wave equations well propagate discontinuites. Besides finitevolume methods compute fluxes across edges in definite directions, increasing the directional effects, whereas the propagation of the physical wave is isotropic. To avoid this, some genuinely multidimensional approaches have been studied (see e.g. Luk´ aˇ cov´ a-Medviˇ dov´ a, Morton and Warnecke [18, 19, 20]). Our goal is to combine classical characteristic methods with multidimensional corrections. These corrections are derived following a method suggested by one of the author (JMG) and has led to [13, 14] and  in the context of gas dynamics (later and independently Abgrall  has used the same type of ideas). It consists in computing the exact solution to the wave equation associated to piece-wise constant initial data on a given mesh.
error estimate is recovered and illustrated with numerical simulations. The method can be extended to other
classical discretizations but the key ingredient is a better approximation rate in the L 2 norm than in the energy
norm, thanks to the Aubin-Nitsche’s trick that is easy for variational approximations. In addition, the degrees of freedom in FVM don’t have the same status as in FEM and the transfer of information from one grid to another must be adapted. The aim of this paper is to propose the adaptation of the NIRB method to FV and to propose the numerical analysis able to recover the classical error estimate with FiniteVolume (FV) schemes.
transmission problem). Of course there are also unknowns defined on the K ∈ M H
and a ∈ V H for the ionic current I ion and the state variables of the ionic models, [X] e ,
[X] i and w.
Around each vertex a ∈ V is built a control volume denoted by A, by joining the centers of gravity and the midpoints of the cells and edges that share a as a common vertex. The idea of the method is to write two finite volumes schemes on the two families of control volumes (K ∈ M and A for a ∈ V). These schemes involve the computation of approximate fluxes on the interfaces between the control volumes. These fluxes rely on a consistent approximation of the gradients of u M and v M .
We study nonlinear hyperbolic conservation laws posed on a differential (n + 1)-manifold with boundary referred to as a spacetime, and in which the “flux” is defined as a flux field of n-forms depending on a parameter (the unknown variable) —a class of equations recently proposed by LeFloch and Okutmustur. Our main result is a proof of the convergence of the finitevolumemethod for weak solutions satisfying suitable entropy inequalities. A main dif- ference with previous work is that we allow here for slices with boundary and, in addition, introduce a new formulation of the finitevolumemethod which involves the notion of total flux functions. Under a natural global hyperbolicity condition on the spacetime and pro- vided the spacetime is foliated by compact slices (with boundary), we establish an existence and uniqueness theory for the initial and boundary value problem, and derive a contraction property in a geometrically natural L 1 -type distance.
Anissa El Keurti and Thomas Rey
Abstract We present a new finitevolumemethod for computing numerical approx- imations of a system of nonlocal transport equation modeling interacting species. This method is based on the work [F. Delarue, F. Lagoutire, N. Vauchelet, Con- vergence analysis of upwind type schemes for the aggregation equation with pointy potential, Ann. Henri. Lebesgue 2019], where the nonlocal continuity equations are treated as conservative transport equations with a nonlocal, nonlinear, rough veloc- ity field. We analyze some properties of the method, and illustrate the results with numerical simulations.
1.2.2. Consideration on the discontinuities of the viscosity. Even for scalar diffusion problems, it is known that such discontinuities in the coefﬁcients imply a consistency defect in the numerical ﬂuxes of usual ﬁnite volume schemes. It is needed to modify the scheme in order to take into account the jumps of the coefﬁcients of the problem and then to recover the optimal ﬁrst order convergence rate. As in the scalar case , we need to introduce a modiﬁed gradient operator (see Deﬁnition 2.5) and ﬁnally deﬁne a modiﬁed approximate viscous stress tensor D η, D N u T (see Deﬁnition 2.7) on each diamond cell. We derive a modiﬁed DDFV
around the vertex of every triangle of the primary mesh.
The rest of this paper is organized as follows. In Section 2, we introduce the chemotaxis model based on realistic biological assumptions, which incorporates the effect of volume-filling mecha- nism and leads to a nonlinear degenerate parabolic system. In Section 3, we derive the control volumefinite element scheme, where an upwind finitevolume scheme is used for the approxima- tion of the convective term, and a standard P1-finite element method is used for the diffusive term. In Section 4, by assuming that the transmissibility coefficients are nonnegative, we prove the maximum principle and give the a priori estimates on the discrete solutions. In Section 5, we show the compactness of the set of discrete solutions by deriving estimates on difference of time and space translates for the approximate solutions. Next, in Section 6, using the Kolmogorov relative compactness theorem, we prove the convergence of a sequence of the approximate so- lutions, and we identify the limits of the discrete solutions as weak solutions of the parabolic system proposed in Section 2. In the last section, we present some numerical simulations to capture the generation of spatial patterns for the volume-filling chemotaxis model with different tensors. These numerical simulations are obtained with our control volumefinite element scheme.
of SWEs which represents the maximum eigenvalue of the Jacobian of the system, is used. Giraldo and Warburton (2005) developed a nodal triangle-based spectral element method for shallow water system on the sphere. Giraldo (2006) developed a triangle-based discontinuous Galerkin method which combines the finite element with finitevolume techniques. In its methodology, the author used the Rusanov flux  for the numerical fluxes where the viscosity coefficient is chosen locally for each Riemann problem and mentioned that more sophisticated Riemann solvers could be used for the study of the shock wave phenomena. Katta et al.(2015) studied two finitevolume methods to solve linear transport problems on the cubed sphere grid system. These numerical methods are based on the central-upwind scheme and high-order reconstructions.