Discrete Transparent Boundary Conditions for Schrödinger-Type Equations

Schmidt, F.; Yevick, David

doi:10.1006/jcph.1997.5675

Cited by 65 publications

(51 citation statements)

References 12 publications

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“…For this, rational interpolatory functions approximating the symbol of the TBC in Fourier variables are used. With these ABC, and if we know a priori the velocity of the solution, it is possible to obtain a very high absorption, comparable to that obtained with TBC [20].…”

Section: Introductionmentioning

confidence: 61%

“…For its numerical approximation, it is necessary to use artificial boundary conditions in order to work in a finite domain [x l , x r ] × R + . When, for x / ∈ (x l , x r ), the initial value u 0 (x) vanishes and the potential V is constant, transparent boundary conditions (TBC) for (1.1) have been studied in [5], [6], [14], [20]. With these TBC, the solution in the computational subdomain coincides with the original solution.…”

Section: Introductionmentioning

confidence: 99%

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Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

Alonso‐Mallo¹,

Reguera²

2003

SIAM J. Numer. Anal.

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Abstract. Recently, some absorbing boundary conditions for Schrödinger-type equations have been studied by Fevens, Jiang and Alonso-Mallo, and Reguera. These conditions make it possible to obtain a very high absorption at the boundary avoiding the nonlocality of transparent boundary conditions. However, the implementations used in the literature, where the boundary condition is chosen in a manual way in accordance with the solution or fixed independently of the solution, are not practical because of the small absorption. In this paper, a new practical adaptive implementation is developed that allows us to obtain automatically a very high absorption.

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Section: Introductionmentioning

confidence: 61%

Section: Introductionmentioning

confidence: 99%

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

Alonso‐Mallo¹,

Reguera²

2003

SIAM J. Numer. Anal.

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“…The importance of the stability problem is well-known for mesh methods with the approximate TBCs of solving time-dependent Schrödinger-like equations in unbounded domains see [1]- [6], [8,9,12], [15]- [17], [19]- [21]. In Part I of this study (see [10]), a new form of the approximate TBCs has been suggested, with a non-local operator S governing properties of the schemes.…”

Section: Introductionmentioning

confidence: 99%

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

Ducomet

Злотник²

2007

Communications in Mathematical Sciences

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Abstract. We continue to consider initial-boundary value problems for a generalized timedependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip. The Crank-Nicolson finite-difference schemes with general approximate transparent boundary conditions (TBCs), including the discrete TBCs, are investigated. We prove unconditional stability in L 2 and in the energy norm with respect to initial data and free terms in the equation and the approximate TBC, in general non-uniform in time, under new suitable conditions (inequalities) on a non-local operator S of the approximate TBC. IntroductionWe continue to consider initial-boundary value problems for a generalized timedependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip closely related to a microscopic description of low-energy nuclear fission dynamics [7,13]. We investigate the Crank-Nicolson finite-difference schemes with general approximate transparent boundary conditions (TBCs) including the discrete TBCs [4,5,12].The importance of the stability problem is well-known for mesh methods with the approximate TBCs of solving time-dependent Schrödinger-like equations in unbounded domains see . In Part I of this study (see [10]), a new form of the approximate TBCs has been suggested, with a non-local operator S governing properties of the schemes. The uniform-in-time stability bounds in L 2 have been proved under suitable condition (inequality) on S, for non-uniform meshes in space and time.In the present Part II, we establish essential further results in this direction. We prove unconditional stability in L 2 and in the energy norm with respect to initial data and free terms in the equation and the approximate TBC, in general non-uniform in time in order to cover broader applications. To this end, we introduce new suitable conditions (inequalities) on S. These inequalities are valid for the operators S ref of the discrete TBCs (ensuring the uniform-in-time stability); we clarify them by considering the corresponding schemes on infinite space meshes. We suggest a trick reducing the derivation of general non-uniform in time bounds to the derivation of simpler uniform

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“…There are many papers [1,2,3,4,5,6,7,8] developing TBCs and studying their difference approximations and stability. However, the obtained TBC through this way is nonlocal in t, thus requiring all the history data in memory.…”

Section: Introductionmentioning

confidence: 99%

Spectral Collocation Technique for Absorbing Boundary Conditions with Increasingly High Order Approximation

Han

2007

Computational Science – ICCS 2007

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Abstract. An efficient treatment is developed for the Schrödinger equation with a class of local absorbing boundary conditions, which are obtained by high order Padé expansions. These boundary conditions are significant in the simulation of open quantum devices. Based on the finite difference approximation in the interior domain, we construct a spectral collocation layer on the cell near the artificial boundary, in which the wave function is approximated by the Chebyshev polynomials. The numerical examples are given by using this strategy with increasingly high order of accuracy up to the ninth order.

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Discrete Transparent Boundary Conditions for Schrödinger-Type Equations

Cited by 65 publications

References 12 publications

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

Discrete Absorbing Boundary Conditions for Schrödinger-Type Equations. Construction and Error Analysis

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

Spectral Collocation Technique for Absorbing Boundary Conditions with Increasingly High Order Approximation

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