The Numerical Computation of the Time Fractional Schrodinger Equation on an Unbounded Domain

A fast and accurate numerical scheme is presented for the computation of the time fractional Schrodinger equation on an unbounded domain. The main idea consists of two parts. First, we use artificial boundary methods to equivalently reformulate the unbounded problem into an initial-boundary value (IBV) problem. Second, we present two numerical schemes for the IBV problem: a direct scheme and a fast scheme. The direct scheme stands for the direct discretization of the Caputo fractional derivative by using the L1-formula. The fast scheme means that the sum-of-exponentials approximation is used to speed up the evaluation of the Caputo fractional derivative. The resulting fast algorithm significantly reduces the storage requirement and the overall computational cost compared to the direct scheme. Furthermore, the corresponding stability analysis and error estimates of two schemes are established, and numerical examples are given to verify the performance of our approach.