TY - JOUR
T1 - Maximum Bound Principle Preserving Integrating Factor Runge–Kutta Methods for Semilinear Parabolic Equations
AU - Ju, Lili
AU - Li, Xiao
AU - Qiao, Zhonghua
AU - Yang, Jiang
N1 - Funding Information:
We are grateful to Professor Chi-Wang Shu of Brown University for many valuable comments. This work is supported by the CAS AMSS-PolyU Joint Laboratory of Applied Mathematics . L. Ju's work is partially supported by US National Science Foundation grant DMS-1818438 and US Department of Energy grant DE-SC0020270 . X. Li's work is partially supported by National Natural Science Foundation of China grant 11801024 . Z. Qiao's work is partially supported by the Hong Kong Research Council GRF grants 15300417 and 15302919 and the Hong Kong Polytechnic University grant G-UAEY . J. Yang's work is supported by National Natural Science Foundation of China grant 11871264 , Natural Science Foundation of Guangdong Province ( 2018A0303130123 ), and NSFC/Hong Kong RGC Joint Research Scheme (NFSC/RGC 11961160718 ).
Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/8/15
Y1 - 2021/8/15
N2 - 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.
AB - 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.
KW - Allen–Cahn equations
KW - High-order numerical methods
KW - Integrating factor Runge–Kutta method
KW - Maximum bound principle
KW - Semilinear parabolic equation
UR - http://www.scopus.com/inward/record.url?scp=85105737941&partnerID=8YFLogxK
U2 - 10.1016/j.jcp.2021.110405
DO - 10.1016/j.jcp.2021.110405
M3 - Journal article
AN - SCOPUS:85105737941
SN - 0021-9991
VL - 439
SP - 1
EP - 18
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 110405
ER -