### Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate

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In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation ${u}_{t}(x,t)+\text{div}F(x,t,u(x,t))=0$ with the initial condition ${u}_{0}\in {L}^{\infty}\left({\mathbb{R}}^{N}\right)$. Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in {L}^{i}nfty({\mathbb{R}}^{N}\times {\mathbb{R}}_{+})$. Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in ${L}_{loc}^{p}({\mathbb{R}}^{N}\times {\mathbb{R}}_{+})$, 1 ≤ ≤ +∞. Furthermore, if ${u}_{0}\in {L}^{\infty}\cap {\text{BV}}_{loc}\left({\mathbb{R}}^{N}\right)$, we show that $u\in {\text{BV}}_{loc}({\mathbb{R}}^{N}\times {\mathbb{R}}_{+})$, which leads to an “${h}^{\frac{1}{4}}$” error estimate between the approximate and the entropy solutions (where defines the size of the...

The aim of this paper is to investigate the stability of boundary layers which appear in numerical solutions of hyperbolic systems of conservation laws in one space dimension on regular meshes. We prove stability under a size condition for Lax Friedrichs type schemes and inconditionnal stability in the scalar case. Examples of unstable boundary layers are also given.

We introduce a finite volume scheme for multi-dimensional drift-diffusion equations. Such equations arise from the theory of semiconductors and are composed of two continuity equations coupled with a Poisson equation. In the case that the continuity equations are non degenerate, we prove the convergence of the scheme and then the existence of solutions to the problem. The key point of the proof relies on the construction of an approximate gradient of the electric potential which allows us to deal...

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