Volume 9, Issue 1
Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations

Liyong Zhu

Adv. Appl. Math. Mech., 9 (2017), pp. 157-172.

Published online: 2018-05

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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 method, explicit scheme, linear splitting, discrete fast Fourier transforms, Allen-Cahn equation.

  • AMS Subject Headings

65M06, 65M22, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-157, author = {Liyong Zhu , }, title = {Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {1}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m1045}, url = {http://global-sci.org/intro/article_detail/aamm/12142.html} }
TY - JOUR T1 - Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations AU - Liyong Zhu , JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 157 EP - 172 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m1045 UR - https://global-sci.org/intro/article_detail/aamm/12142.html KW - Exponential Runge-Kutta method, explicit scheme, linear splitting, discrete fast Fourier transforms, Allen-Cahn equation. AB -

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.

Liyong Zhu. (2020). Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations. Advances in Applied Mathematics and Mechanics. 9 (1). 157-172. doi:10.4208/aamm.2015.m1045
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