A High-Efficient Algorithm for Parabolic Problems with Time-Dependent Coefficients (Advances in Applied Mathematics and Mechanics, 2018)
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-ExtrapolationAlgorithm (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 parabolicproblems, 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 linearcombination of exact matrix values in level $L, L+s, L+2s$ to predict matrix values inthe following $s−1$ levels, and the coefficients of the linear combination is deduced bythe 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 fullmatrix decreases by a factor $1/s$. Last, the MTEA is analyzed in detail and the effectivenessof new method is verified by numerical experiments.