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Volume 9, Issue 2
A High-Efficient Algorithm for Parabolic Problems with Time-Dependent Coefficients

Chuanmiao Chen, Xiangqi Wang & Hongling Hu

Adv. Appl. Math. Mech., 9 (2017), pp. 501-514.

Published online: 2018-05

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  • Abstract

A high-efficient algorithm to solve Crank-Nicolson scheme for variable coefficient parabolic problems is studied in this paper, which consists of the Function Time-Extrapolation Algorithm (FTEA) and Matrix Time-Extrapolation Algorithm (MTEA). First, FTEA takes a linear combination of previous $l$ level solutions ($U^{n,0}$=$∑^l_{i=1}$$a_i$$U^{n−i}$) as good initial value of $U^n$ (see Time-extrapolation algorithm (TEA) for linear parabolic problems, J. Comput. Math., 32(2) (2014), pp. 183–194), so that Conjugate Gradient (CG)-iteration counts decrease to 1/3∼1/4 of direct CG. Second, MTEA uses a linear combination of exact matrix values in level $L, L+s, L+2s$ to predict matrix values in the following $s−1$ levels, and the coefficients of the linear combination is deduced by the quadric interpolation formula, then fully recalculate the matrix values at time level $L+3s$, and continue like this iteratively. Therefore, the number of computing the full matrix decreases by a factor $1/s$. Last, the MTEA is analyzed in detail and the effectiveness of new method is verified by numerical experiments.

  • Keywords

Crank-Nicolson scheme, Time-Extrapolation, CG-iteration, variable coefficient parabolic.

  • AMS Subject Headings

65M06, 65M12, 65B05, 65N22

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-501, author = {Chuanmiao and Chen and and 19539 and and Chuanmiao Chen and Xiangqi and Wang and and 19540 and and Xiangqi Wang and Hongling and Hu and and 19541 and and Hongling Hu}, title = {A High-Efficient Algorithm for Parabolic Problems with Time-Dependent Coefficients}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {2}, pages = {501--514}, abstract = {

A high-efficient algorithm to solve Crank-Nicolson scheme for variable coefficient parabolic problems is studied in this paper, which consists of the Function Time-Extrapolation Algorithm (FTEA) and Matrix Time-Extrapolation Algorithm (MTEA). First, FTEA takes a linear combination of previous $l$ level solutions ($U^{n,0}$=$∑^l_{i=1}$$a_i$$U^{n−i}$) as good initial value of $U^n$ (see Time-extrapolation algorithm (TEA) for linear parabolic problems, J. Comput. Math., 32(2) (2014), pp. 183–194), so that Conjugate Gradient (CG)-iteration counts decrease to 1/3∼1/4 of direct CG. Second, MTEA uses a linear combination of exact matrix values in level $L, L+s, L+2s$ to predict matrix values in the following $s−1$ levels, and the coefficients of the linear combination is deduced by the quadric interpolation formula, then fully recalculate the matrix values at time level $L+3s$, and continue like this iteratively. Therefore, the number of computing the full matrix decreases by a factor $1/s$. Last, the MTEA is analyzed in detail and the effectiveness of new method is verified by numerical experiments.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m1281}, url = {http://global-sci.org/intro/article_detail/aamm/12161.html} }
TY - JOUR T1 - A High-Efficient Algorithm for Parabolic Problems with Time-Dependent Coefficients AU - Chen , Chuanmiao AU - Wang , Xiangqi AU - Hu , Hongling JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 501 EP - 514 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m1281 UR - https://global-sci.org/intro/article_detail/aamm/12161.html KW - Crank-Nicolson scheme, Time-Extrapolation, CG-iteration, variable coefficient parabolic. AB -

A high-efficient algorithm to solve Crank-Nicolson scheme for variable coefficient parabolic problems is studied in this paper, which consists of the Function Time-Extrapolation Algorithm (FTEA) and Matrix Time-Extrapolation Algorithm (MTEA). First, FTEA takes a linear combination of previous $l$ level solutions ($U^{n,0}$=$∑^l_{i=1}$$a_i$$U^{n−i}$) as good initial value of $U^n$ (see Time-extrapolation algorithm (TEA) for linear parabolic problems, J. Comput. Math., 32(2) (2014), pp. 183–194), so that Conjugate Gradient (CG)-iteration counts decrease to 1/3∼1/4 of direct CG. Second, MTEA uses a linear combination of exact matrix values in level $L, L+s, L+2s$ to predict matrix values in the following $s−1$ levels, and the coefficients of the linear combination is deduced by the quadric interpolation formula, then fully recalculate the matrix values at time level $L+3s$, and continue like this iteratively. Therefore, the number of computing the full matrix decreases by a factor $1/s$. Last, the MTEA is analyzed in detail and the effectiveness of new method is verified by numerical experiments.

Chuanmiao Chen, Xiangqi Wang & Hongling Hu. (2020). A High-Efficient Algorithm for Parabolic Problems with Time-Dependent Coefficients. Advances in Applied Mathematics and Mechanics. 9 (2). 501-514. doi:10.4208/aamm.2015.m1281
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