Using IIR filters to build high-order finite difference schemes for the unsteady Schrödinger equation

Abstract

High-order finite difference schemes for the time-dependent Schrödinger equation are investigated. Digital signal processing methods allowed proving the conservativeness of high-order finite difference schemes for the unsteady Schrödinger equation. The eighth-order scheme coefficients were found with the help of the proved theoretical results. The conditions for equivalence between the eighth-order finite difference scheme and the scheme in the form of a cascade of allpass first-order filters were found. The numerical analysis of the proposed scheme was made. It was shown that the high-order finite difference schemes gave better results on solving the linear Schrödinger equations comparing to the well-known fourthorder scheme on the six-point stencil, however, the high-order schemes in couple with the second-order splitting algorithm to the nonlinear Schrödinger equation do not lead to a radical improvement in the quality of numerical results. Practical issues implementing the proposed numerical technique are considered. The obtained results can be used to construct efficient solvers for linear and nonlinear Schrödinger-type equations by applying the splitting schemes of adequate accuracy order.