Abstract
Two stable finite difference schemes, with the second-order Crank-Nicolson/Adams-Bashforth method in time and the second-order finite difference approach in space, are proposed for solving the phase-field Allen-Cahn equation with a small parameter perturbation and strong nonlinearity. First, we design a finite difference scheme with conditional energy stability for the Allen-Cahn equation, which preserves the energy-decreasing property, and prove its error estimate. Second, by introducing an artificial stability term, we establish an unconditional energy-stable finite difference scheme for the Allen-Cahn equation, discuss its unconditional energy stability, and obtain its error estimate. Finally, numerical examples are presented to verify all of the theoretical results for the proposed schemes.
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