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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

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 [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].

  • Keywords

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

  • AMS Subject Headings

65M06, 65M12, 65M25, 35F10.

  • Copyright

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 [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].

}, 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, Immersed interface upwind scheme, Piecewise constant coefficients, Error estimate, 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 [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].

Xin Wen and Shi Jin. (2008). 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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