| Issue |
ESAIM: M2AN
Volume 50, Number 2, March-April 2016
|
|
|---|---|---|
| Page(s) | 499 - 539 | |
| DOI | https://doi.org/10.1051/m2an/2015057 | |
| Published online | 11 March 2016 | |
On the convergence rate of finite difference methods for degenerate convection-diffusion equations in several space dimensions
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
kennethk@math.uio.no; nilshr@math.uio.no; erlenbs@math.uio.no
Received: 27 January 2015
Revised: 10 July 2015
We analyze upwind difference methods for strongly degenerate convection-diffusion equations in several spatial dimensions. We prove that the local L1-error between the exact and numerical solutions is 𝓞(Δx2 / (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 [K.H. Karlsen, N.H. Risebro E.B. Storrøsten, Math. Comput. 83 (2014) 2717–2762], in which the one-dimensional case was examined using the Kružkov−Carrillo entropy framework.
Mathematics Subject Classification: 65M06 / 65M15 / 35K65 / 35L65
Key words: Degenerate convection-diffusion equations / entropy conditions / finite difference methods / error estimates
© EDP Sciences, SMAI 2016
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