Fast algorithms of compact scheme for solving parabolic equations and their application

Abstract

Based on existing work on compact scheme, particularly utilizing the Crank-Nicolson scheme for time derivatives and compact difference schemes for spatial derivatives in solving linear parabolic equations, we propose a fast algorithm of the scheme to solve the systems for the first time. Given that the resulting coefficient matrices of the scheme are diagonalizable, we transform the matrix-vector equations into a diagonal component-wise system, utilizing modified discrete cosine transform (MDCT), discrete sine transform (DST), and discrete Fourier transform (DFT) to optimize CPU time and reduce storage requirements. Moreover, the algorithmic technique facilitates a novelty and simple convergence demonstration strategy in the discrete maximum norm that is easily extendable to high-dimensional linear cases. The computational framework is also extendable to three-dimensional (3D) linear case and semi-linear case. Numerical experiments are given to support our findings.