Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations

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. The MBP plays a crucial role in understanding the physical meaning...

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Veröffentlicht in:Journal of computational physics Jg. 439; S. 110405
Hauptverfasser: Ju, Lili, Li, Xiao, Qiao, Zhonghua, Yang, Jiang
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cambridge Elsevier Inc 15.08.2021
Elsevier Science Ltd
Elsevier
Schlagworte:
Algorithms
Allen-Cahn equations
Boundary conditions
Computational physics
Discrete systems
Gradient flow
High-order numerical methods
Integrating factor Runge-Kutta method
Mathematical models
MATHEMATICS AND COMPUTING
Maximum bound principle
Numerical methods
Runge-Kutta method
Semilinear parabolic equation
Time dependence
Time integration
Allen–Cahn equations
Integrating factor Runge–Kutta method
Semilinear parabolic equation
High-order numerical methods
Maximum bound principle
ISSN:0021-9991, 1090-2716
Online-Zugang:Volltext
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Zusammenfassung: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. The MBP plays a crucial role in understanding the physical meaning and the wellposedness 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.
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National Science Foundation (NSF)
Hong Kong Research Council
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR). Scientific Discovery through Advanced Computing (SciDAC)
National Natural Science Foundation of China (NSFC)
USDOE Office of Science (SC), Biological and Environmental Research (BER). Earth and Environmental Systems Science Division
SC0020270; DMS-1818438; 11801024; 15300417; 15302919; 11871264
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2021.110405