Stabilized leapfrog scheme preserving the maximum bound principle for the generalized Allen–Cahn equation

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Abstract

This paper addresses the numerical method for the generalized Allen–Cahn equation featuring nonlinear mobility and a convection term. We propose a linear second–order finite difference scheme that adheres to the discrete maximum bound principle (MBP). The scheme is discretized using the leapfrog finite difference approach, incorporating a stabilized term in time, an upwind scheme for the convection term, and a central–difference scheme for the diffusion term. It is demonstrated that the discrete MBP holds under reasonable constraints on both the time step size and the coefficient of the stabilized term. Additionally, we provide an L^∞–error estimate for our proposed scheme. Several numerical experiments are conducted to validate our theoretical findings.

Keywords

generalized Allen–Cahn equation / finite difference method / maximum bound principle / error estimates

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