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Volume 36, Issue 4
Relaxation Exponential Runge-Kutta Methods and Their Applications to Semilinear Dissipative/Conservative Systems

Dongfang Li, Xiaoxi Li & Jiang Yang

DOI: 10.4208/cicp.OA-2023-0241

Commun. Comput. Phys., 36 (2024), pp. 908-942.

Published online: 2024-10

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  • Abstract

This paper presents a family of novel relaxation exponential Runge-Kutta methods for semilinear partial differential equations with dissipative/conservative energy. The novel methods are developed by using the relaxation idea and adding a well-designed governing equation to explicit exponential Runge-Kutta methods. It is shown that the proposed methods can be of high-order accuracy and energy-stable/conserving with mild time step restrictions. In contrast, the previous explicit exponential-type methods are not energy-conserving. Several numerical experiments on KdV equations, Schrödinger equations and Navier-Stokes equations are carried out to illustrate the effectiveness and high efficiency of the methods.

  • Keywords

Explicit exponential Runge-Kutta methods, relaxation technique, high-order accuracy, structure-preserving property.

  • AMS Subject Headings

65L04, 65L06, 65M12

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-36-908, author = {Li , DongfangLi , Xiaoxi and Yang , Jiang}, title = {Relaxation Exponential Runge-Kutta Methods and Their Applications to Semilinear Dissipative/Conservative Systems}, journal = {Communications in Computational Physics}, year = {2024}, volume = {36}, number = {4}, pages = {908--942}, abstract = {

This paper presents a family of novel relaxation exponential Runge-Kutta methods for semilinear partial differential equations with dissipative/conservative energy. The novel methods are developed by using the relaxation idea and adding a well-designed governing equation to explicit exponential Runge-Kutta methods. It is shown that the proposed methods can be of high-order accuracy and energy-stable/conserving with mild time step restrictions. In contrast, the previous explicit exponential-type methods are not energy-conserving. Several numerical experiments on KdV equations, Schrödinger equations and Navier-Stokes equations are carried out to illustrate the effectiveness and high efficiency of the methods.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2023-0241}, url = {http://global-sci.org/intro/article_detail/cicp/23481.html} }
TY - JOUR T1 - Relaxation Exponential Runge-Kutta Methods and Their Applications to Semilinear Dissipative/Conservative Systems AU - Li , Dongfang AU - Li , Xiaoxi AU - Yang , Jiang JO - Communications in Computational Physics VL - 4 SP - 908 EP - 942 PY - 2024 DA - 2024/10 SN - 36 DO - http://doi.org/10.4208/cicp.OA-2023-0241 UR - https://global-sci.org/intro/article_detail/cicp/23481.html KW - Explicit exponential Runge-Kutta methods, relaxation technique, high-order accuracy, structure-preserving property. AB -

This paper presents a family of novel relaxation exponential Runge-Kutta methods for semilinear partial differential equations with dissipative/conservative energy. The novel methods are developed by using the relaxation idea and adding a well-designed governing equation to explicit exponential Runge-Kutta methods. It is shown that the proposed methods can be of high-order accuracy and energy-stable/conserving with mild time step restrictions. In contrast, the previous explicit exponential-type methods are not energy-conserving. Several numerical experiments on KdV equations, Schrödinger equations and Navier-Stokes equations are carried out to illustrate the effectiveness and high efficiency of the methods.

Li , DongfangLi , Xiaoxi and Yang , Jiang. (2024). Relaxation Exponential Runge-Kutta Methods and Their Applications to Semilinear Dissipative/Conservative Systems. Communications in Computational Physics. 36 (4). 908-942. doi:10.4208/cicp.OA-2023-0241
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