On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions

KH Karlsen, NH Risebro, EB Storrøsten - … : Mathematical Modelling and …, 2016 - numdam.org
ESAIM: Mathematical Modelling and Numerical Analysis, 2016numdam.org
We analyze upwind difference methods for strongly degenerate convection-diffusion
equations in several spatial dimensions. We prove that the local L 1-error between the exact
and numerical solutions is O (Δx 2/(19+ d)), where d is the spatial dimension and Δx is the
grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof
makes effective use of specific kinetic formulations of the difference method and the
convection-diffusion equation. This paper is a continuation of [KH Karlsen, NH Risebro EB …
Abstract
We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local L 1-error between the exact and numerical solutions is O (Δx 2/(19+ d)), where d is the spatial dimension and Δx is the grid size. The error estimate is robust with respect to vanishing diffusion effects. The proof makes effective use of specific kinetic formulations of the difference method and the convection-diffusion equation. This paper is a continuation of [KH Karlsen, NH Risebro EB Storrøsten, Math. Comput. 83 (2014) 2717–2762], in which the one-dimensional case was examined using the Kruzkov− Carrillo entropy framework.
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