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

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.

65M06, 65M12, 65B05, 65N22

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@Article{AAMM-9-501, author = {Chuanmiao Chen , and Xiangqi Wang , and Hu , Hongling}, 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 - Chuanmiao Chen , AU - Xiangqi Wang , 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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