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An error estimate for the finite difference approximation to degenerate convection–diffusion equations

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Abstract

We consider semi-discrete first-order finite difference schemes for a nonlinear degenerate convection–diffusion equations in one space dimension, and prove an L 1 error estimate. Precisely, we show that the \({L^1_{\rm{loc}}}\) difference between the approximate solution and the unique entropy solution converges at a rate \({\mathcal{O}(\Delta x^{1/11})}\) , where \({\Delta x}\) is the spatial mesh size. If the diffusion is linear, we get the convergence rate \({\mathcal{O}(\Delta x^{1/2})}\) , the point being that the \({\mathcal{O}}\) is independent of the size of the diffusion.

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Authors and Affiliations

  1. Centre of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, Oslo, 0316, Norway

    K. H. Karlsen, U. Koley & N. H. Risebro

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  1. K. H. Karlsen
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  2. U. Koley
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  3. N. H. Risebro
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Correspondence to N. H. Risebro.

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Karlsen, K.H., Koley, U. & Risebro, N.H. An error estimate for the finite difference approximation to degenerate convection–diffusion equations. Numer. Math. 121, 367–395 (2012). https://doi.org/10.1007/s00211-011-0433-9

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  • DOI: https://doi.org/10.1007/s00211-011-0433-9

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