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Volume 27, Issue 6
On a Moving Mesh Method for Solving Partial Integro-Differential Equations

Jingtang Ma, Yingjun Jiang & Kaili Xiang

DOI: 10.4208/jcm.2009.09-m2852

J. Comp. Math., 27 (2009), pp. 713-728.

Published online: 2009-12

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  • Abstract

This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a piecewise quadratic polynomial in space and a piecewise linear function in time. Then, an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.

  • Keywords

Partial integro-differential equations, Moving mesh methods, Stability and Convergence.

  • AMS Subject Headings

65R20, 65N50, 65M50, 65M20, 65M15, 65M10.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-713, author = {Jingtang Ma, Yingjun Jiang and Kaili Xiang}, title = {On a Moving Mesh Method for Solving Partial Integro-Differential Equations}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {6}, pages = {713--728}, abstract = {

This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a piecewise quadratic polynomial in space and a piecewise linear function in time. Then, an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m2852}, url = {http://global-sci.org/intro/article_detail/jcm/8599.html} }
TY - JOUR T1 - On a Moving Mesh Method for Solving Partial Integro-Differential Equations AU - Jingtang Ma, Yingjun Jiang & Kaili Xiang JO - Journal of Computational Mathematics VL - 6 SP - 713 EP - 728 PY - 2009 DA - 2009/12 SN - 27 DO - http://doi.org/10.4208/jcm.2009.09-m2852 UR - https://global-sci.org/intro/article_detail/jcm/8599.html KW - Partial integro-differential equations, Moving mesh methods, Stability and Convergence. AB -

This paper develops and analyzes a moving mesh finite difference method for solving partial integro-differential equations. First, the time-dependent mapping of the coordinate transformation is approximated by a piecewise quadratic polynomial in space and a piecewise linear function in time. Then, an efficient method to discretize the memory term of the equation is designed using the moving mesh approach. In each time slice, a simple piecewise constant approximation of the integrand is used, and thus a quadrature is constructed for the memory term. The central finite difference scheme for space and the backward Euler scheme for time are used. The paper proves that the accumulation of the quadrature error is uniformly bounded and that the convergence of the method is second order in space and first order in time. Numerical experiments are carried out to confirm the theoretical predictions.

Jingtang Ma, Yingjun Jiang and Kaili Xiang. (2009). On a Moving Mesh Method for Solving Partial Integro-Differential Equations. Journal of Computational Mathematics. 27 (6). 713-728. doi:10.4208/jcm.2009.09-m2852
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