Fast High-Order Compact Exponential Time Differencing Runge–Kutta Methods for Second-Order Semilinear Parabolic Equations

In this paper we propose fast high-order numerical methods for solving a class of second-order semilinear parabolic equations in regular domains. The proposed methods are explicit in nature, and use exponential time differencing and Runge–Kutta approximations in combination with a linear splitting technique to achieve accurate and stable time integration. A two-step compact difference scheme is employed for spatial discretization to obtain fourth-order accuracy and make use of FFT-based fast calculations. Such methods can be applied to problems with stiff nonlinearities and boundary conditions of Dirichlet or periodic types. Linear stability analysis and various numerical experiments are also presented to demonstrate accuracy and stability of the proposed methods.