Fourth-order finite difference scheme and efficient algorithm for nonlinear fractional Schrödinger equations

Abstract

Abstract To improve the computing efficiency, a fourth-order difference scheme is proposed and a fast algorithm is designed to simulate the nonlinear fractional Schrödinger (FNLS) equation oriented from the fractional quantum mechanics. The numerical analysis and experiments conducted in this article show that the proposed difference scheme has the optimal second-order and fourth-order convergence rates in time and space respectively, reduces its computation cost to O(MlogM) $\mathcal{O}(M\log M)$, and recognizes accurately its physical feature of FNLS such as the mass balance.

Keywords

• Fractional Schrödinger equation
• Fourth-order difference scheme
• Fast algorithm
• Mass balance
• Numerical analysis