Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations (Advances in Applied Mathematics and Mechanics, 2018)
In this paper we develop explicit fast exponential Runge-Kutta methods forthe numerical solutions of a class of parabolic equations. By incorporating the linearsplitting technique into the explicit exponential Runge-Kutta schemes, we are able togreatly improve the numerical stability. The proposed numerical methods could befast implemented through use of decompositions of compact spatial difference operatorson a regular mesh together with discrete fast Fourier transform techniques. Theexponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximationswith variable time step sizes, as well as applied to stiff nonlinearity andboundary conditions of different types. Linear stabilities of the proposed schemes andtheir comparison with other schemes are presented. We also numerically demonstrateaccuracy, stability and robustness of the proposed method through some typical modelproblems.