A bound-preserving scheme for the Allen–Cahn equation

Abstract

We propose and analyze a bound-preserving scheme for the Allen–Cahn equation. The key idea is to apply a bound-preserving nonlinear stabilization technique to the implicit Euler time-stepping method coupled with the continuous finite element method. To our best knowledge, this is the first scheme which theoretically preserves the maximum principle and has an error estimate that is optimal in the $L^2\left(H^1\right)$-seminorm and with a polynomial dependence on ${ϵ}^{-1}$ at the same time. The proof of the error estimate combines a nonlinear Ritz projection together with a special Grönwall inequality. Numerical experiments are conducted to compare the performance of our scheme with a bound-preserving operator-splitting scheme.