An Energy Stable TFPM-Based Petrov-Galerkin Scheme for Solving the Allen-Cahn Equation

Abstract

An energy stable tailored finite point method based Petrov-Galerkin scheme to solve Allen-Cahn equation is proposed. In time discretization, we present both first-order and second-order semi-discrete schemes based on stabilized and convex-splitting techniques, which satisfy unconditional energy stability. We prove the maximum bound preserving principle for first-order schemes. Due to nonlinearity, the well-posedness of weak formulations based on semi-discrete schemes are demonstrated. As the nature of singularly perturbation in semi-discrete level remains when $ε$ is extremely small, we establish a specified Petrov-Galerkin scheme which leads to a unified way for space discretization. To this end, we set up nonlinear solvers which are proved to be stable and convergent. Then we construct our Petrov-Galerkin scheme, which is built upon problem-dependent test function space. The stability and second-order convergence of this scheme are rigorously proved in one dimension. In order to compute test functions, specialized TFPM schemes are incorporated into the scheme. Numerical experiments show the accuracy, efficiency, and the good performance of the method on uniform meshes even when mesh size $h$ is much larger than $ε$.