Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for variable coefficients linear and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [12] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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“…In this goal, we first have to define the E-quasi hyperbolic, elliptic and glancing zones [7], with E = (1, 2). Definition 3.1.…”

Section: Remark About the Convergence Of Cswr Methods In Real Timementioning

confidence: 99%

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

2018

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“…[41,42,43]. Let us finally remark that applications to generalized Schrödinger equations could also be developed by adapting the methods developed in [2,44]. …”

Section: Resultsmentioning

confidence: 99%

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

Klein

Antoine

Besse

et al. 2011

Commun. comput. phys.

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We propose a hierarchy of novel absorbing boundary conditions for the one-dimensional stationary Schrödinger equation with general (linear and nonlinear) potential. The accuracy of the new absorbing boundary conditions is investigated numerically for the computation of energies and ground-states for linear and nonlinear Schrödinger equations. It turns out that these absorbing boundary conditions and their variants lead to a higher accuracy than the usual Dirichlet boundary condition. Finally, we give the extension of these ABCs to N-dimensional stationary Schrödinger equations.

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“…The natural and mathematically correct types of artificial boundary conditions are approximate TBCs (either non-local or local). It is well-known that their construction is not trivial since such approximate TBCs should ensure both stability of the resulting method in the bounded domain and small numerical reflections from the artificial boundary, see [1]- [4], [7,9,10,13], [15]- [17], [19,20].…”

Section: Introductionmentioning

confidence: 99%

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Ducomet¹,

Zlotnik²

2006

Communications in Mathematical Sciences

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Abstract. We consider initial-boundary value problems for a generalized time-dependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip. For Crank-Nicolson finite-difference schemes, we suggest an alternative coupling to approximate transparent boundary conditions and present a condition ensuring unconditional stability. In the case of discrete transparent boundary conditions, we revisit the statement and the proof of stability together with the derivation of the conditions.

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Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions

Cited by 89 publications

References 24 publications

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves

Absorbing Boundary Conditions for Solving N-Dimensional Stationary Schrödinger Equations with Unbounded Potentials and Nonlinearities

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

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