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 - <p style="text-align: justify;">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.</p>