Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations
Abstract
A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemesmore »
- Authors:
-
- Univ. of South Carolina, Columbia, SC (United States)
- Hong Kong Polytechnic Univ., Kowloon (Hong Kong)
- Southern Univ. of Science and Technology, Shenzhen (China)
- Publication Date:
- Research Org.:
- Univ. of South Carolina, Columbia, SC (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC); USDOE Office of Science (SC), Biological and Environmental Research (BER). Earth and Environmental Systems Science Division; National Science Foundation (NSF); National Natural Science Foundation of China (NSFC); Hong Kong Research Council
- OSTI Identifier:
- 1785013
- Grant/Contract Number:
- SC0020270; DMS-1818438; 11801024; 15300417; 15302919; 11871264
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 439; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; Maximum bound principle; Integrating factor Runge-Kutta method; Semilinear parabolic equation; High-order numerical methods; Allen-Cahn equations
Citation Formats
Ju, Lili, Li, Xiao, Qiao, Zhonghua, and Yang, Jiang. Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations. United States: N. p., 2021.
Web. doi:10.1016/j.jcp.2021.110405.
Ju, Lili, Li, Xiao, Qiao, Zhonghua, & Yang, Jiang. Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations. United States. https://doi.org/10.1016/j.jcp.2021.110405
Ju, Lili, Li, Xiao, Qiao, Zhonghua, and Yang, Jiang. Wed .
"Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations". United States. https://doi.org/10.1016/j.jcp.2021.110405. https://www.osti.gov/servlets/purl/1785013.
@article{osti_1785013,
title = {Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations},
author = {Ju, Lili and Li, Xiao and Qiao, Zhonghua and Yang, Jiang},
abstractNote = {A large class of semilinear parabolic equations satisfy the maximum bound principle (MBP) in the sense that the time-dependent solution preserves for any time a uniform pointwise bound imposed by its initial and boundary conditions. Here, the MBP plays a crucial role in understanding the physical meaning and the well-posedness of the mathematical model. Investigation on numerical algorithms with preservation of the MBP has attracted increasingly attentions in recent years, especially for the temporal discretizations, since the violation of MBP may lead to nonphysical solutions or even blow-ups of the algorithms. In this paper, we study high-order MBP-preserving time integration schemes by means of the integrating factor Runge-Kutta (IFRK) method. Beginning with the space-discrete system of semilinear parabolic equations, we present the IFRK method in general form and derive the sufficient conditions for the method to preserve the MBP. In particular, we show that the classic four-stage, fourth-order IFRK scheme is MBP preserving for some typical semilinear systems although not strong stability preserving, which can be instantly applied to the Allen-Cahn type of equations. To our best knowledge, this is the first time to present a fourth-order linear numerical method preserving the MBP. In addition, convergence of these numerical schemes is proved theoretically and verified numerically, as well as their efficiency by simulations of 2D and 3D long-time evolutional behaviors. Numerical experiments are also carried out for a model which is not a typical gradient flow as the Allen-Cahn type of equations.},
doi = {10.1016/j.jcp.2021.110405},
journal = {Journal of Computational Physics},
number = ,
volume = 439,
place = {United States},
year = {Wed May 05 00:00:00 EDT 2021},
month = {Wed May 05 00:00:00 EDT 2021}
}
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