Efficient, non-iterative, and second-order accurate numerical algorithms for the anisotropic Allen–Cahn Equation with precise nonlocal mass conservation

Abstract

We propose two efficient, non-iterative, and second-order accurate algorithms to solving the anisotropic Allen–Cahn equation with the nonlocal mass conservation. The first scheme is the stabilized-SAV approach which combines the SAV approach with the stabilization technique, in which three linear stabilization terms are added to enhance the stability and keep the required accuracy while using large time steps. The scheme not only can conserve the mass precisely but also is very easy-to-implement and non-iterative where one only needs to solve three decoupled fourth-order biharmonic equations with constant coefficients at each time step. We further prove the scheme is unconditionally energy stable rigorously. The second scheme is based on the linear stabilization approach where all nonlinear terms are treated in an explicit way, therefore the scheme is quite efficient and stable that allows for large time steps in computations. For both schemes, we present a number of 2D and 3D numerical simulations to show stability and accuracy. • The anisotropic Cahn-Hilliard model is reformulated by using nonlocal Allen-Cahn equation. • Two efficient, non-iterative, and second-order algorithms are developed to preserve the mass. • Three linear stabilization terms are added to enhance the stability that allows large time step. • Numerous 2D and 3D numerical simulations to demonstrate the stability and accuracy.