Numerical Analysis and Computation of the Finite Volume Element Method for the Nonlinear Coupled Time-Fractional Schrödinger Equations

Abstract

In this article, our aim is to consider an efficient finite volume element method combined with the L2−1σ formula for solving the coupled Schrödinger equations with nonlinear terms and time-fractional derivative terms. We design the fully discrete scheme, where the space direction is approximated using the finite volume element method and the time direction is discretized making use of the L2−1σ formula. We then prove the stability for the fully discrete scheme, and derive the optimal convergence result, from which one can see that our scheme has second-order accuracy in both the temporal and spatial directions. We carry out numerical experiments with different examples to verify the optimal convergence result.

Keywords

• coupled time-fractional Schrödinger equations
• finite volume element method
• <i>L</i>2 − 1_<i>σ</i> formula
• optimal error estimates