Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

Abstract

We consider numerical methods for solving the fractional-in-space Allen–Cahn equation which contains small perturbation parameters and strong nonlinearity. A standard fully discretized scheme for this equation is considered, namely, using the conventional second-order Crank–Nicolson scheme in time and the second-order central difference approach in space. For the resulting nonlinear scheme, we propose a nonlinear iteration algorithm, whose unique solvability and convergence can be proved. The nonlinear iteration can avoid inverting a dense matrix with only \(\mathcal {O}(N\log N)\) computation complexity. One major contribution of this work is to show that the numerical solutions satisfy discrete maximum principle under reasonable time step constraint. Based on the maximum stability, the nonlinear energy stability for the fully discrete scheme is established, and the corresponding error estimates are investigated. Numerical experiments are performed to verify the theoretical results.