Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations

doi:10.1007/s42967-020-00118-x

Abstract

Abstract: In this paper, we study the classical Allen-Cahn equations and investigate the maximumprinciple-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fuid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the AllenCahn equation may yield thin interface layer, and nonuniform meshes might be useful in the numerical solutions. Therefore, we apply the local discontinuous Galerkin (LDG) method due to its fexibility on h-p adaptivity and complex geometry. However, the MPP LDG methods require slope limiters, then the energy stability may not be easy to obtain. In this paper, we only discuss the MPP technique and use numerical experiments to demonstrate the energy decay property. Moreover, due to the stif source given in the equation, we use the conservative modifed exponential Runge-Kutta methods and thus can use relatively large time step sizes. Thanks to the conservative time integration, the bounds of the unknown function will not decay. Numerical experiments will be given to demonstrate the good performance of the MPP LDG scheme.

Key words: Maximum-principle-preserving, Local discontinuous Galerkin methods, Allen-Cahn equation, Conservative exponential integrations

CLC Number:

65M12
65M60

Jie Du, Eric Chung, Yang Yang. Maximum-Principle-Preserving Local Discontinuous Galerkin Methods for Allen-Cahn Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(1): 353-379.

TrendMD

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