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

Ju，Lili1; Li，Xiao2; Qiao，Zhonghua2; Yang，Jiang3

2021-08-15

10.1016/j.jcp.2021.110405

JOURNAL OF COMPUTATIONAL PHYSICS   影响因子和分区

0021-9991

1090-2716

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.

Allen–Cahn equations High-order numerical methods Integrating factor Runge–Kutta method Maximum bound principle Semilinear parabolic equation

SCI ; EI

英语

其他

WOS:000663421700002

20212010363912

Numerical methods ; Partial differential equations

Calculus:921.2 ; Numerical Methods:921.6

PHYSICS

2-s2.0-85105737941

Scopus

1.Department of Mathematics,University of South Carolina,Columbia,29208,United States
2.Department of Applied Mathematics,The Hong Kong Polytechnic University,Kowloon,Hung Hom,Hong Kong
3.Department of Mathematics,SUSTech International Center for Mathematics,Southern University of Science and Technology,Shenzhen,518055,China

Ju，Lili,Li，Xiao,Qiao，Zhonghua,等. Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations[J]. JOURNAL OF COMPUTATIONAL PHYSICS,2021,439.

Ju，Lili,Li，Xiao,Qiao，Zhonghua,&Yang，Jiang.(2021).Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations.JOURNAL OF COMPUTATIONAL PHYSICS,439.

Ju，Lili,et al."Maximum bound principle preserving integrating factor Runge–Kutta methods for semilinear parabolic equations".JOURNAL OF COMPUTATIONAL PHYSICS 439(2021).