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$ L^1$ error estimates for difference approximations of degenerate convection-diffusion equations


Authors: K. H. Karlsen, N. H. Risebro and E. B. Storrøsten
Journal: Math. Comp. 83 (2014), 2717-2762
MSC (2010): Primary 65M06, 65M15; Secondary 35K65, 35L65
DOI: https://doi.org/10.1090/S0025-5718-2014-02818-4
Published electronically: March 28, 2014
MathSciNet review: 3246807
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Abstract: We analyze monotone finite difference schemes for strongly 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 and exact solutions is $ \mathcal {O}(\Delta x^{1/3})$, where $ \Delta x$ is the spatial grid parameter. This result should be compared with the classical $ \mathcal {O}(\Delta x^{1/2})$ error estimate for conservation laws (Kuznecov, 1976), and a recent estimate of $ \mathcal {O}(\Delta x^{1/11})$ for degenerate convection-diffusion equations (Karlsen, Koley, Risebro 2012).


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Additional Information

K. H. Karlsen
Affiliation: Center of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email: kennethk@math.uio.no

N. H. Risebro
Affiliation: Center of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email: nilshr@math.uio.no

E. B. Storrøsten
Affiliation: Center of Mathematics for Applications (CMA), University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
Email: erlenbs@math.uio.no, erlenbs@gmail.com

DOI: https://doi.org/10.1090/S0025-5718-2014-02818-4
Keywords: Degenerate convection-diffusion equations, entropy conditions, finite difference schemes, error estimates
Received by editor(s): May 4, 2012
Received by editor(s) in revised form: January 25, 2013
Published electronically: March 28, 2014
Article copyright: © Copyright 2014 American Mathematical Society