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Volume 9, Issue 3
Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions

K. H. Karlsen & J. D. Towers

Adv. Appl. Math. Mech., 9 (2017), pp. 515-542.

Published online: 2018-05

[An open-access article; the PDF is free to any online user.]

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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.

  • AMS Subject Headings

35K65, 35L65, 65M06, 65M08, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-515, author = {Karlsen , K. H. and Towers , J. D.}, title = {Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {3}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2016.m-s1}, url = {http://global-sci.org/intro/article_detail/aamm/12162.html} }
TY - JOUR T1 - Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions AU - Karlsen , K. H. AU - Towers , J. D. JO - Advances in Applied Mathematics and Mechanics VL - 3 SP - 515 EP - 542 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2016.m-s1 UR - https://global-sci.org/intro/article_detail/aamm/12162.html KW - Degenerate parabolic equation, scalar conservation law, zero-flux boundary condition, monotone scheme, convergence. AB -

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.

K. H. Karlsen & J. D. Towers. (2020). Convergence of Monotone Schemes for Conservation Laws with Zero-Flux Boundary Conditions. Advances in Applied Mathematics and Mechanics. 9 (3). 515-542. doi:10.4208/aamm.2016.m-s1
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