Linearized compact difference methods combined with Richardson extrapolation for nonlinear delay Sobolev equations

Abstract

Delay Sobolev equations (DSEs) are a class of important models in fluid mechanics, thermodynamics and the other related fields. For solving this class of equations, in this paper, linearized compact difference methods (LCDMs) for one- and two-dimensional problems of DSEs are suggested. The solvability and convergence of the methods are analyzed and it is proved under some appropriate conditions that the methods are convergent of order two in time and order four in space. In order to improve the computational accuracy of LCDMs in time, we introduce the Richardson extrapolation technique, which leads to the improved LCDMs can reach the fourth-order accuracy in both time and space. Finally, with several numerical experiments, the theoretical accuracy and computational effectiveness of the proposed methods are further testified.