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Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions

Part of: Parabolic equations and systems Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  17 January 2017

K. H. Karlsen  and
J. D. Towers
K. H. Karlsen*
Affiliation:
Department of Mathematics, University of Oslo, P.O. Box 1053, Blindern, N–0316 Oslo, Norway
J. D. Towers*
Affiliation:
MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516, USA
*
*Corresponding author. Email:kennethk@math.uio.no (K. H. Karlsen), john.towers@cox.net (J. D. Towers)
*Corresponding author. Email:kennethk@math.uio.no (K. H. Karlsen), john.towers@cox.net (J. D. Towers)
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Abstract

We consider a scalar conservation law with zero-flux boundary conditions imposed on the boundary of a rectangular multidimensional domain. We study monotone schemes applied to this problem. For the Godunov version of the scheme, we simply set the boundary flux equal to zero. For other monotone schemes, we additionally apply a simple modification to the numerical flux. We show that the approximate solutions produced by these schemes converge to the unique entropy solution, in the sense of [7], of the conservation law. Our convergence result relies on a BV bound on the approximate numerical solution. In addition, we show that a certain functional that is closely related to the total variation is nonincreasing from one time level to the next. We extend our scheme to handle degenerate convection-diffusion equations and for the one-dimensional case we prove convergence to the unique entropy solution.

Keywords

Degenerate parabolic equationscalar conservation lawzero-flux boundary conditionmonotone schemeconvergence

MSC classification

Secondary: 35K65: Degenerate parabolic equations 35L65: Conservation laws 65M06: Finite difference methods 65M08: Finite volume methods 65M12: Stability and convergence of numerical methods
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 3 , June 2017 , pp. 515 - 542
Copyright
Copyright © Global-Science Press 2017 

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