Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient |
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Authors: | Karlsen K H; Risebro N H; Towers J D |
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Institution: |
1 Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N5008 Bergen, Norway 2 Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N0316 Oslo, Norway 3 MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516, USA
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Abstract: | We analyse approximate solutions generated by an upwind differencescheme (of EngquistOsher type) for nonlinear degenerateparabolic convectiondiffusion equations where the nonlinearconvective flux function has a discontinuous coefficient (x)and the diffusion function A(u) is allowed to be strongly degenerate(the pure hyperbolic case is included in our setup). The mainproblem is obtaining a uniform bound on the total variationof the difference approximation u, which is a manifestationof resonance. To circumvent this analytical problem, we constructa singular mapping (, ·) such that the total variationof the transformed variable z = (, u) can be bounded uniformlyin . This establishes strong L1 compactness of z and, since(, ·) is invertible, also u. Our singular mapping isnovel in that it incorporates a contribution from the diffusionfunction A(u). We then show that the limit of a converging sequenceof difference approximations is a weak solution as well as satisfyinga Krukov-type entropy inequality. We prove that the diffusionfunction A(u) is Hölder continuous, implying that the constructedweak solution u is continuous in those regions where the diffusionis nondegenerate. Finally, some numerical experiments are presentedand discussed. |
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Keywords: | degenerate convection diffusion equation discontinuous coefficient weak solution finite difference scheme convergence entropy condition |
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