Linearized fast time-stepping schemes for time–space fractional Schrödinger equations

Abstract

This paper proposes a linearized fast high-order time-stepping scheme to solve the time–space fractional Schrödinger equations. The time approximation is done by using the fast L2-1σ formula on nonuniform meshes. The spatial discretization is done by using Fourier spectral methods. Optimal error estimates of the numerical method are obtained unconditionally. Such unconditional convergence results are proved directly under the assumption f∈C1(R) by the fractional Sobolev inequalities and boundedness of the numerical solutions. In contrast, the previous unconditionally convergence results are usually proved by using the error splitting argument under the assumption f∈C3(R) and the help of some additional systems. Numerical experiments are given to confirm the theoretical results.