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We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce Lperror estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce $L^p$ error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete $H^1$ finite volume space. We actually prove the convergence of the scheme in a discrete $H^1$ norm, with an error estimate of order $O(h)$ (on meshes of size $h$).

We study a finite volume method, used to approximate the solution of the linear two dimensional convection diffusion equation, with mixed Dirichlet and Neumann boundary conditions, on Cartesian meshes refined by an automatic technique (which leads to meshes with hanging nodes). We propose an analysis through a discrete variational approach, in a discrete H1finite volume space. We actually prove the convergence of the scheme in a discrete H1norm, with an error estimate of order O(h)(on meshes of size h).

In this paper, a class of cell centered finite volume schemes, on general unstructured meshes, for a linear convection-diffusion problem, is studied. The convection and the diffusion are respectively approximated by means of an upwind scheme and the so called diamond cell method [4]. Our main result is an error estimate of order h, assuming only the W2,p(for p>2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in [12]. Some new difficulties arise here, due to the weak regularity of the solution, and the necessity to approximate the entire gradient, and not only its normal component, as in [12].

• A domain-decomposition technique is designed to obtain scalable very high-order finite volumes schemes. • This technique is applied to simulations of the monodomain model in electrocardiology, which is a reaction-diffusion equation coupled to a system of ODEs. • The high-order FV scheme follows a MOOD strategy with 2 sets of stencils chosen such as to restore a relevent scalability. • Numerical experiments prove the efficiency of an OpenMP implementation tested on CPUs (max 24 threads) and KNL nodes (max 256 threads) of this technique (scalability, preserved order of accuracy and accuracy). In this paper, a domain decomposition technique for a very high-order finite volumes scheme is proposed. The objective is to obtain an efficient way to perform numerical simulations in cardiac electrophysiology. The aim is to extend a very high-order numerical scheme previously designed, where large stencils are used for polynomial reconstructions. Therefore, a particular attention has to be paid to maintain the scalability in parallel. Here, we propose to constrain the stencils inside the subdomains or their first layer of neighbors. The method is shown to remain accurate and to scale perfectly up to the level where there are not enough cells in the subdomains. Hence, these high-order schemes are proved to be efficient tools to perform realistic simulations in cardiac electrophysiology. [ABSTRACT FROM AUTHOR]