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

Nissen, Anna; Kreiss, Gunilla; Gerritsen, Margot

doi:10.1007/s10915-012-9586-7

Cited by 19 publications

(41 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The super-convergence of the sixth order accurate scheme is also observed when solving the Schrödinger equation [17,18]. To test the Neumann problem, we again use (43) as the analytical solution and impose the Neumann boundary condition at two boundaries x = 0 and x = 1.…”

Section: The One Dimensional Wave Equationmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

Wang

Kreiss

2016

J Sci Comput

View full text Add to dashboard Cite

When using a finite difference method to solve a time dependent partial differential equation, the truncation error is often larger at a few grid points near a boundary or grid interface than in the interior. In computations, the observed convergence rate is often higher than the order of the large truncation error. In this paper, we develop techniques for analyzing this phenomenon, and particularly consider the second order wave equation. The equation is discretized by a finite difference operator satisfying a summation by parts property, and the boundary and grid interface conditions are imposed weakly by the simultaneous approximation term method. It is well-known that if the semi-discretized wave equation satisfies the determinant condition, that is the boundary system in Laplace space is nonsingular for all Re(s) ≥ 0, two orders are gained from the large truncation error localized at a few grid points. By performing a normal mode analysis, we show that many common discretizations do not satisfy the determinant condition at s = 0. We then carefully analyze the error equation to determine the gain in the convergence rate. The result shows that stability does not automatically imply a gain of two orders in the convergence rate. The precise gain can be lower than, equal to or higher than two orders, depending on the boundary condition and numerical boundary treatment. The accuracy analysis is verified by numerical experiments, and very good agreement is obtained.

show abstract

Section: The One Dimensional Wave Equationmentioning

confidence: 99%

“…In [17], the Schrödinger equation with a grid interface is considered and is shown to be of this type. Analysis in Laplace space is performed and yields sharper error estimates than the 1/2 order gain obtained by applying the energy method to the error equation in physical space.…”

Section: Introductionmentioning

confidence: 99%

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

Wang

Kreiss

2016

J Sci Comput

View full text Add to dashboard Cite

show abstract

“…Multi-block schemes and in particular interface procedures that use Summationby-Parts (SBP) operators together with the Simultaneous Approximation Term (SAT) technique [2], have previously been investigated in terms of conservation, stability and accuracy [3,5,6,10,11,27,28,29,30]. The focus of the SBP-SAT methodology has been, so far, mostly on time-independent spatial domains with a notable exception being [13].…”

Section: Introductionmentioning

confidence: 99%

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Nikkar

Nordstrm

2017

Journal of Computational Physics

View full text Add to dashboard Cite

We introduce an interface/coupling procedure for hyperbolic problems posed on time-dependent curved multi-domains. First, we transform the problem from Cartesian to boundary-conforming curvilinear coordinates and apply the energy method to derive well-posed and conservative interface conditions. Next, we discretize the problem in space and time by employing finite difference operators that satisfy a summation-by-parts rule. The interface condition is imposed weakly using a penalty formulation. We show how to formulate the penalty operators such that the coupling procedure is automatically adjusted to the movements and deformations of the interface, while both stability and conservation conditions are respected.The developed techniques are illustrated by performing numerical experiments on the linearized Euler equations and the Maxwell equations. The results corroborate the stability and accuracy of the fully discrete approximations.

show abstract

“…The SBP-SAT technique has been extended to include curvilinear coordinate transforms [32,42], multi-block couplings [6,31,7,34,23,28], artificial dissipation operators [26,8], and has been applied to numerous applications where it has proven to be robust. See for example [44,25,14,16].…”

Section: Introductionmentioning

confidence: 99%

Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations

Berg

Nordström

2014

Journal of Computational Physics

View full text Add to dashboard Cite

In this paper we derive new farfield boundary conditions for the time-dependent Navier-Stokes and Euler equations in two space dimensions. The new boundary conditions are derived by simultaneously considering well-posedess of both the primal and dual problems. We moreover require that the boundary conditions for the primal and dual Navier-Stokes equations converge to well-posed boundary conditions for the primal and dual Euler equations.We perform computations with a high-order finite difference scheme on summationby-parts form with the new boundary conditions imposed weakly by the simultaneous approximation term. We prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non-linear integral functionals become superconvergent.

show abstract

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

Cited by 19 publications

References 19 publications

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

Convergence of Summation-by-Parts Finite Difference Methods for the Wave Equation

A fully discrete, stable and conservative summation-by-parts formulation for deforming interfaces

Duality based boundary conditions and dual consistent finite difference discretizations of the Navier–Stokes and Euler equations

Contact Info

Product

Resources

About