Year: 2017
Author: K. H. Karlsen, J. D. Towers
Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 3 : pp. 515–542
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.2016.m-s1
Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 3 : pp. 515–542
Published online: 2017-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Degenerate parabolic equation scalar conservation law zero-flux boundary condition monotone scheme convergence.