A linear doubly stabilized Crank-Nicolson scheme for the Allen-Cahn equation with a general mobility

TL;DR

This paper introduces a linear, second-order, unconditionally stable numerical scheme for the Allen-Cahn equation with general mobility, combining Crank-Nicolson and finite difference methods with stabilization for improved accuracy and stability.

Contribution

It develops a novel linear scheme that unconditionally preserves the maximum bound principle and provides rigorous error and energy stability analyses for both constant and variable mobility cases.

Findings

The scheme unconditionally preserves the maximum bound principle.

Error estimates are derived in H^1 and L^∞ norms.

Numerical experiments confirm theoretical results and demonstrate efficiency.

Abstract

In this paper, a linear second order numerical scheme is developed and investigated for the Allen-Cahn equation with a general positive mobility. In particular, our fully discrete scheme is mainly constructed based on the Crank-Nicolson formula for temporal discretization and the central finite difference method for spatial approximation, and two extra stabilizing terms are also introduced for the purpose of improving numerical stability. The proposed scheme is shown to unconditionally preserve the maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stability simultaneously. With the help of uniform boundedness of the numerical solutions due to MBP, we then successfully derive $H^{1}$-norm and $L^{\infty}$-norm error estimates for the Allen-Cahn equation with a constant and a variable mobility, respectively. Moreover, the energy stability of the proposed scheme is also obtained in the sense that the discrete free energy is uniformly bounded by the one at the initial time plus a {\color{black}constant}. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the performance of the proposed scheme with a time adaptive strategy.