Crank-Nicolson Legendre spectral approximation for space-fractional Allen-Cahn equation

Abstract

In this article, we consider spectral methods for solving the initial-boundary value problem of the space fractional-order Allen-Cahn equation. A fully discrete scheme based on the modified Crank-Nicolson scheme in time and the Legendre spectral method in space is established. The existence and uniqueness of the fully discrete scheme are derived, and the stability and convergence analysis of the fully discrete scheme are proved rigorously. By constructing a fractional duality argument, the corresponding optimal error estimates in L2 and Hα norm are derived, respectively. Also, numerical experiments are performed to support the theoretical results.