An unconditional stable compact fourth-order finite difference scheme for three dimensional Allen–Cahn equation

Abstract

In this paper, we present an unconditional stable linear high-order finite difference scheme for three dimensional Allen–Cahn equation. This scheme, which is based on a backward differentiation scheme combined with a fourth-order compact finite difference formula, is second order accurate in time and fourth order accurate in space. A linearly stabilized splitting scheme is used to remove the restriction of time step. We prove the unconditional stability of our proposed method in analysis. A fast and efficient linear multigrid solver is employed to solve the resulting discrete system. We perform various numerical experiments to confirm the high-order accuracy, unconditional stability and efficiency of our proposed method. In particular, we show two applications of our proposed method: triply-periodic minimal surface and volume inpainting.