Anna Nissen

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

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 to include stability and accuracy at nonconforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions. The accuracy of the grid interface coupling is investigated using normal mode analysis for operators of 2nd and 4th order formal interior accuracy. We show that full accuracy is retained for the 2nd and 4th order operators. The accuracy results are extended to 6th and 8th order operators by numerical simulations, in which case two orders of accuracy is gained with respect to the lower order approximation close to the interface.

High Order Stable Finite Difference Methods for the Schrödinger Equation

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.

Stable Difference Methods for Block-Oriented Adaptive Grids

In this paper, we present a block-oriented scheme for adaptive mesh refinement based on summation-by-parts (SBP) finite difference methods and simultaneous-approximation-term (SAT) interface treatment. Since the order of accuracy at SBP–SAT grid interfaces is lower compared to that of the interior stencils, we strive at using the interior stencils across block-boundaries whenever possible. We devise a stable treatment of SBP-FD junction points, i.e. points where interfaces with different boundary treatment meet. This leads to stable discretizations for more flexible grid configurations within the SBP–SAT framework, with a reduced number of SBP–SAT interfaces. Both first and second derivatives are considered in the analysis. Even though the stencil order is locally reduced close to numerical interfaces and corner points, numerical simulations show that the locally reduced accuracy does not severely reduce the accuracy of the time propagated numerical solution. Moreover, we explain how to organize the grid and how to automatically adapt the mesh, aiming at problems of many variables. Examples of adaptive grids are demonstrated for the simulation of the time-dependent Schrödinger equation and for the advection equation.

A perfectly matched layer applied to a reactive scattering problem

The perfectly matched layer (PML) technique is applied to a reactive scattering problem for accurate domain truncation. A two-dimensional model for dissociative adsorbtion and associative desorption of H(2) from a flat surface is considered, using a finite difference spatial discretization and the Arnoldi method for time-propagation. We compare the performance of the PML to that of a monomial complex absorbing potential, a transmission-free complex absorbing potential, and to exterior complex scaling. In particular, the reflection properties due to the numerical treatment are investigated. We conclude that the PML is accurate and efficient, especially if high accuracy is of significance. Moreover, we demonstrate that the errors from the PML can be controlled at a desired accuracy, enabling efficient numerical simulations.

Error Control for Simulations of a Dissociative Quantum System

We present a framework for solving the Schrodinger equation modeling the interaction of a dissociative quantum system with a laser field. A perfectly matched layer (PML) is used to handle non-reflecting boundaries and the Schrodinger equation is discretized with high-order finite differences in space and an h, p-adaptive Magnus–Arnoldi propagator in time. We use a posteriori error estimation theory to control the global error of the numerical discretization. The parameters of the PML are chosen to meet the same error tolerance. We apply our framework to the IBr molecule, for which numerical experiments show that the total error can be controlled efficiently. Moreover, we provide numerical evidence that the Magnus–Arnoldi solver outperforms the implicit Crank–Nicolson scheme by far.

Convergence of finite difference methods for the wave equation in two space dimensions

When using a finite difference method to solve an initial-boundaryvalue problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.