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Front Tracking and Operator Splitting for Nonlinear Degenerate Convection-Diffusion Equations

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Parallel Solution of Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 120))

Abstract

We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convection—diffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.

Karlsen has been supported by VISTA, a research cooperation between the Norwegian Academy of Science and Letters and Den norske stats oljeselskap a.s. (Statoil).

Lie has been supported by the Research Council of Norway under grant 100555/410.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008, Bergen, Norway

    S. Evje & K. Hvistendahl Karlsen

  2. Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7034, Trondheim, Norway

    K. A. Lie

  3. Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N-0316, Oslo, Norway

    N. H. Risebro

Authors
  1. S. Evje
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  2. K. Hvistendahl Karlsen
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  3. K. A. Lie
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  4. N. H. Risebro
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Editor information

Editors and Affiliations

  1. Institutt for Informatikk Hoyteknologisenteret, University of Bergen, Bergen, N-5020, Norway

    Petter Bjørstad

  2. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Mitchell Luskin

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Evje, S., Karlsen, K.H., Lie, K.A., Risebro, N.H. (2000). Front Tracking and Operator Splitting for Nonlinear Degenerate Convection-Diffusion Equations. In: Bjørstad, P., Luskin, M. (eds) Parallel Solution of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 120. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1176-1_9

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  • DOI: https://doi.org/10.1007/978-1-4612-1176-1_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7034-8

  • Online ISBN: 978-1-4612-1176-1

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