High Order Stable Finite Difference Methods for the Schrödinger Equation

Abstract

In this paper we extend the Summation-by-parts-simultaneous approximation term (SBP-SAT) technique to the Schrödinger equation. Stability estimates are derived and the accuracy of numerical approximations of interior order 2m, m = 1, 2, 3, are analyzed in the case of Dirichlet boundary conditions. We show that a boundary closure of the numerical approximations of order m lead to global accuracy of order m + 2. The results are supported by numerical simulations.

Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schrödinger Equation

“…In this paper we extend the results from [8] in the following ways. We demonstrate how to apply the SAT technique to yield stability at grid interfaces in one dimension.…”

“…In this paper we extend the results from our earlier work on stable boundary closures for the Schrödinger equation using the summation-by-parts-simultaneous approximation term (SBP-SAT) method [8] to include stability and accuracy at non-conforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions.…”

“…In an earlier work, we presented a framework for analyzing accuracy of stable boundary closures, and applied it to the Schrödinger equation [8]. In this paper we extend the results from [8] in the following ways.…”

“…Therefore, we aim at non-decaying energy estimates for the interface problem, as in the stability estimates in [8].…”

“…The stability results are then extended to nonconforming grid blocks in two dimensions using the interpolation operators constructed in [7]. We also consider the accuracy of stable grid couplings by the approach established in [8]. The methodology is based on a separation of interior truncation error and interface truncation error.…”