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

Kenneth Hvistendahl Karlsen; Nils Henrik Risebro; Erlend Briseid Storrøsten

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

Analysis of PDEs 2015-09-07 v4

Authors: Kenneth Hvistendahl Karlsen , Nils Henrik Risebro , Erlend Briseid Storrøsten

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 $\mathcal{O}(\Delta x^{2/(19+d)})$ , where $d$ is the spatial dimension and $\Delta x$ 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.

Keywords

degenerate convection-diffusion equations finite difference methods convergence rate error estimate upwind schemes kinetic formulations spatial dimensions

Cite

@article{arxiv.1411.4538,
  title  = {On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions},
  author = {Kenneth Hvistendahl Karlsen and Nils Henrik Risebro and Erlend Briseid Storrøsten},
  journal= {arXiv preprint arXiv:1411.4538},
  year   = {2015}
}