Volume 32, Issue 2
Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem

Hongling Hu, Chuanmiao Chen & Kejia Pan

J. Comp. Math., 32 (2014), pp. 183-194.

Published online: 2014-04

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  • Abstract

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic problems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method (MG) is greatly reduced. Numerical experiments compare the efficiency of Direct Conjugate Gradient Method (DCG) and Extrapolation Cascadic Multigrid Method (EXCMG). Last, we propose a Time- Extrapolation Algorithm (TEA), which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG*s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

  • Keywords

Parabolic problem Crank-Nicolson scheme Error estimates Time-extrapolation algorithm CG-iteration

  • AMS Subject Headings

65B05 65M06 65M12 65N22.

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COPYRIGHT: © Global Science Press

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@Article{JCM-32-183, author = {Hongling Hu, Chuanmiao Chen and Kejia Pan}, title = {Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem}, journal = {Journal of Computational Mathematics}, year = {2014}, volume = {32}, number = {2}, pages = {183--194}, abstract = {

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic problems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method (MG) is greatly reduced. Numerical experiments compare the efficiency of Direct Conjugate Gradient Method (DCG) and Extrapolation Cascadic Multigrid Method (EXCMG). Last, we propose a Time- Extrapolation Algorithm (TEA), which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG*s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1310-FE1}, url = {http://global-sci.org/intro/article_detail/jcm/9877.html} }
TY - JOUR T1 - Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem AU - Hongling Hu, Chuanmiao Chen & Kejia Pan JO - Journal of Computational Mathematics VL - 2 SP - 183 EP - 194 PY - 2014 DA - 2014/04 SN - 32 DO - http://doi.org/10.4208/jcm.1310-FE1 UR - https://global-sci.org/intro/article_detail/jcm/9877.html KW - Parabolic problem KW - Crank-Nicolson scheme KW - Error estimates KW - Time-extrapolation algorithm KW - CG-iteration AB -

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic problems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method (MG) is greatly reduced. Numerical experiments compare the efficiency of Direct Conjugate Gradient Method (DCG) and Extrapolation Cascadic Multigrid Method (EXCMG). Last, we propose a Time- Extrapolation Algorithm (TEA), which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG*s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

Hongling Hu, Chuanmiao Chen & Kejia Pan. (1970). Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem. Journal of Computational Mathematics. 32 (2). 183-194. doi:10.4208/jcm.1310-FE1
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