Volume 26, Issue 1
Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients I: L^1-Error Estimates

Xin Wen & Shi Jin

DOI:

J. Comp. Math., 26 (2008), pp. 1-22

Published online: 2008-02

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  • Abstract

We study the $L^1$-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in $L^1$-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order $L^1$-error bounds with explicit coefficients following a technique used in \cite{TT}. We also use some inequalities on binomial coefficients proved in a consecutive paper \cite{WJ2}.

  • Keywords

Linear advection equations Immersed interface upwind scheme Piecewise constant coefficients Error estimate Half order error bound

  • AMS Subject Headings

65M06 65M12 65M25 35F10.

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COPYRIGHT: © Global Science Press

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@Article{JCM-26-1, author = {Xin Wen and Shi Jin}, title = {Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients I: L^1-Error Estimates}, journal = {Journal of Computational Mathematics}, year = {2008}, volume = {26}, number = {1}, pages = {1--22}, abstract = { We study the $L^1$-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in $L^1$-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order $L^1$-error bounds with explicit coefficients following a technique used in \cite{TT}. We also use some inequalities on binomial coefficients proved in a consecutive paper \cite{WJ2}.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10363.html} }
TY - JOUR T1 - Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients I: L^1-Error Estimates AU - Xin Wen & Shi Jin JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 22 PY - 2008 DA - 2008/02 SN - 26 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10363.html KW - Linear advection equations KW - Immersed interface upwind scheme KW - Piecewise constant coefficients KW - Error estimate KW - Half order error bound AB - We study the $L^1$-error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in $L^1$-norm to the differential equation with the corresponding interface condition. We derive the one-halfth order $L^1$-error bounds with explicit coefficients following a technique used in \cite{TT}. We also use some inequalities on binomial coefficients proved in a consecutive paper \cite{WJ2}.
Xin Wen & Shi Jin. (1970). Convergence of an Immersed Interface Upwind Scheme for Linear Advection Equations with Piecewise Constant Coefficients I: L^1-Error Estimates. Journal of Computational Mathematics. 26 (1). 1-22. doi:
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