Abstract
In this paper, we use finite difference methods for solving the Allen–Cahn equation that contains small perturbation parameters and strong nonlinearity. We consider a linearized second-order three-level scheme in time and a second-order finite difference approach in space, and establish discrete boundedness stability in maximum norm: if the initial data are bounded by 1, then the numerical solutions in later times can also be bounded uniformly by 1. It is shown that the main result can be obtained under certain restrictions on the time step.
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