Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg–Landau equation with fractional Laplacian in unbounded domain

Abstract

This paper proposes a fast and efficient spectral-Galerkin method for the nonlinear complex Ginzburg–Landau equation involving the fractional Laplacian in Rd. By employing the Fourier-like bi-orthogonal mapped Chebyshev function as basis functions, the fractional Laplacian can be fully diagonalized. Then for the resulting diagonalized semi-discrete system, an exponential time differencing scheme is proposed for the temporal discretization. The obtained method can be fast implemented and has second order accuracy in time and algebraical accuracy in space. One- and two-dimensional numerical examples are tested to validate the accuracy and efficiency of the proposed method.