Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

Year:    2017

Author:    Liyong Zhu

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 1 : pp. 157–172

Abstract

In this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.2015.m1045

Advances in Applied Mathematics and Mechanics, Vol. 9 (2017), Iss. 1 : pp. 157–172

Published online:    2017-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    16

Keywords:    Exponential Runge-Kutta method explicit scheme linear splitting discrete fast Fourier transforms Allen-Cahn equation.

Author Details

Liyong Zhu