A fast compact time integrator method for a family of general order semilinear evolution equations

Abstract In this paper we develop a fast compact time integrator method for numerically solving a family of general order semilinear evolution equations in regular domains. The spatial discretization is carried out by a fourth-order accurate compact difference scheme in which fast Fourier transform can be utilized for efficient implementation. The resulting semi-discretized problem consists of a system of ordinary differential equations whose solution can be explicitly expressed in term of time integrators, and a desired numerical method is then obtained by further adopting multistep approximations of the nonlinear terms based on the solution formula. Linear stability analysis is performed for the method for second-order in time evolution equations. Extensive numerical experiments with applications are also presented to demonstrate efficiency, accuracy, and stability of the proposed method in practice.