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# L¹ Error Estimates for Difference Approximations of Degenerate Convection-Diffusion Equations

Abstract: We analyze a class of semi-discrete monotone difference schemes for degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the $L^1$ error between the approximate solutions and the unique entropy solution is $\mathcal{O}(∆x^{1/3})$, where $∆x$ denotes the spatial discretization parameter. This result should be compared with the classical $\mathcal{O}(∆x^{1/2})$ result for conservation laws [20], and a very recent error estimate of $\mathcal{O}(∆x^{1/11})$ for degenerate convection-diffusion equations [18].

References
[18] K. H. Karlsen, U. Koley, and N. H. Risebro. An error estimate for the finite difference approximation to degenerate convection-diffusion equations. Numer. Math., 2012.
[20] N. N. Kuznetsov. The accuracy of certain approximate methods for the computation of weak solutions of a first order quasilinear equation. U.S.S.R. Computational Math. and Math. Phys., 16(6):105–119, 1976.
Paper:
Available as PDF (384 Kbytes).
Author(s):
Kenneth H. Karlsen
Nils Henrik Risebro
Erlend B. Storrøsten
Submitted by:
; 2012-05-04.