Adv. Appl. Math. Mech., 9 (2017), pp. 501-514.
Published online: 2018-05
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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} }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.