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Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Computer aspects of numerical algorithms

Published online by Cambridge University Press:  11 October 2016

Liyong Zhu
Liyong Zhu*
Affiliation:
School of Mathematics and Systems Science, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China
*
*Corresponding author. Email:liyongzhu@buaa.edu.cn (L. Y. Zhu)
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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.

Keywords

Exponential Runge-Kutta methodexplicit schemelinear splittingdiscrete fast Fourier transformsAllen-Cahn equation

MSC classification

Secondary: 65M06: Finite difference methods 65M22: Solution of discretized equations 65Y20: Complexity and performance of numerical algorithms
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 157 - 172
Copyright
Copyright © Global-Science Press 2017 

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