A Linear Doubly Stabilized Crank-Nicolson Scheme for the Allen–Cahn Equation with a General Mobility (Advances in Applied Mathematics and Mechanics, 2024)
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 formulafor 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 preservethe maximum bound principle (MBP) under mild restrictions on the stabilization parameters, which is of practical importance for achieving good accuracy and stabilitysimultaneously. With the help of uniform boundedness of the numerical solutions dueto MBP, we then successfully derive $H^1$-norm and $L^∞$-norm error estimates for theAllen–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 constant.Finally, some numerical experiments are carried out to verify the theoretical resultsand illustrate the performance of the proposed scheme with a time adaptive strategy.