Maximum-principle-preserving local discontinuous Galerkin methods for Allen-Cahn equations,

Jie Du Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China; Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China Eric Chung Department of Mathematics, The Chinese University of Hong Kong, Hong Kong SAR, China Yang Yang Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

Numerical Analysis and Scientific Computing mathscidoc:2205.25021

Communications on Applied Mathematics and Computation, 4, 353-379, 2022.4
In this paper, we study the classical Allen-Cahn equations and investigate the maximum-principle-preserving (MPP) techniques. The Allen-Cahn equation has been widely used in mathematical models for problems in materials science and fluid dynamics. It enjoys the energy stability and the maximum-principle. Moreover, it is well known that the Allen-Cahn 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 flexibility 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 stiff source given in the equation, we use the conservative modified 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.
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  title={Maximum-principle-preserving local discontinuous Galerkin methods for Allen-Cahn equations,},
  author={Jie Du, Eric Chung, and Yang Yang},
  booktitle={Communications on Applied Mathematics and Computation},
Jie Du, Eric Chung, and Yang Yang. Maximum-principle-preserving local discontinuous Galerkin methods for Allen-Cahn equations,. 2022. Vol. 4. In Communications on Applied Mathematics and Computation. pp.353-379.
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