Fast algorithm for nonlocal Allen–Cahn equation with scalar auxiliary variable approach

Abstract

Numerical analysis is presented for the nonlocal Allen–Cahn equation, which contains spatial nonlocal operator and time-fractional derivative. By employing the spatial quadrature-based finite difference method and the nonuniform L1 formula jointed with the scalar auxiliary variable (SAV) approach in temporal discretization, a nonuniform numerical scheme is established. The nonlinear solver can be transformed into linear one effectively due to the SAV approach. And, the proposed scheme is proven to be energy stable by use of the positive definiteness of the kernel function. Moreover, the fast algorithm based on the nonuniform L1 formula is applied in the numerical example to improving computational efficiency. Finally, the numerical results demonstrate the temporal convergence of numerical scheme, energy property, comparisons with the nonlocal cases and local cases and maximum principle of the numerical solution.