Analysis of a compact multi-step ADI method for linear parabolic equation

Abstract

ABSTRACTAs we know, second-order backward differentiation formula (BDF2) is L-stable scheme, which can damp unwanted finite oscillations, while Crank-Nicolson (C-N) method is only A-stable scheme and usually causes numerical oscillations. Thus, BDF2 may be more popular than C-N. However, up to now, most of studies focus on the error analysis of the compact alternating direction implicit (ADI) method on the basis of C-N method for temporal discretization in -norm. This paper is devoted to the convergence analysis of a compact multi-step ADI method used to solve a two-dimensional (2D) parabolic equation in -, - and -norms. Besides, the existence of the ADI solution is also analyzed in detail. A class of Richardson extrapolation algorithms are established to improve computational efficiency. By introducing a transformation, our algorithms are easily generalized to the solution of convection-diffusion problems with constant coefficients. Numerical results confirm the performance of our algorithms and support theoretical analysis.