Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation* *Project supported by the National Key R&D Program of China (Grant No. 2020YFA0709800), the National Natural Science Foundation of China (Grant Nos. 11901577, 11971481, 12071481, and 12001539), the Natural Science Foundation of Hunan, China (Grant Nos. S2017JJQNJJ0764 and 2020JJ5652), the fund from Hunan Provincial Key Laboratory of

Abstract

We propose a novel energy dissipative method for the Allen–Cahn equation on nonuniform grids. For spatial discretization, the classical central difference method is utilized, while the average vector field method is applied for time discretization. Compared with the average vector field method on the uniform mesh, the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation. This is due to the moving mesh method, which can concentrate the grid points more densely where the solution changes drastically. Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.