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Volume 25, Issue 6
Approximation, Stability and Fast Evaluation of Exact Artificial Boundary Condition for the One-Dimensional Heat Equation

Chunxiong Zheng

J. Comp. Math., 25 (2007), pp. 730-745.

Published online: 2007-12

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  • Abstract

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

  • Keywords

Heat equation, Artificial boundary conditions, Fast evaluation, Unbounded domains.

  • AMS Subject Headings

65N12, 65M12, 26A33.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-25-730, author = {Chunxiong Zheng}, title = {Approximation, Stability and Fast Evaluation of Exact Artificial Boundary Condition for the One-Dimensional Heat Equation}, journal = {Journal of Computational Mathematics}, year = {2007}, volume = {25}, number = {6}, pages = {730--745}, abstract = {

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8726.html} }
TY - JOUR T1 - Approximation, Stability and Fast Evaluation of Exact Artificial Boundary Condition for the One-Dimensional Heat Equation AU - Chunxiong Zheng JO - Journal of Computational Mathematics VL - 6 SP - 730 EP - 745 PY - 2007 DA - 2007/12 SN - 25 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8726.html KW - Heat equation, Artificial boundary conditions, Fast evaluation, Unbounded domains. AB -

In this paper we consider the numerical solution of the one-dimensional heat equation on unbounded domains. First an exact semi-discrete artificial boundary condition is derived by discretizing the time variable with the Crank-Nicolson method. The semi-discretized heat equation equipped with this boundary condition is then proved to be unconditionally stable, and its solution is shown to have second-order accuracy. In order to reduce the computational cost, we develop a new fast evaluation method for the convolution operation involved in the exact semi-discrete artificial boundary condition. A great advantage of this method is that the unconditional stability held by the semi-discretized heat equation is preserved. An error estimate is also given to show the dependence of numerical errors on the time step and the approximation accuracy of the convolution kernel. Finally, a simple numerical example is presented to validate the theoretical results.

Chunxiong Zheng. (2007). Approximation, Stability and Fast Evaluation of Exact Artificial Boundary Condition for the One-Dimensional Heat Equation. Journal of Computational Mathematics. 25 (6). 730-745. doi:
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