On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Ducomet, Bernard; Злотник, А. А.; Zlotnik, Ilya

doi:10.3934/krm.2009.2.151

Cited by 19 publications

(48 citation statements)

References 11 publications

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“…Propositions 1-5 and Lemma 1 in the quite similar cases of the semi-infinite Π = (0, ∞) (n = 1) and Π = (0, ∞) × (0, X 2 ) (n = 2) were proved respectively in [4] and [12,15] (see also [16]), where families of finite-difference schemes with space averages depending on a parameter θ were treated covering, in particular, the linear and bilinear FEMs (for θ = 1 6 ). For the presented improvement in formulas (12), see also [14].…”

Section: The Ibvp and Numerical Methods To Solve Itmentioning

confidence: 82%

Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

Злотник

2015

Finite Difference Methods,Theory and Applications

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We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded n-dimensional parallelepiped (n ≥ 1). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error estimates O(τ 2 +|h| 2 ) uniformly in time in L 2 space norm, for n ≥ 1, and mesh H 1 space norm, for 1 ≤ n ≤ 3 (a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

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Section: The Ibvp and Numerical Methods To Solve Itmentioning

confidence: 82%

Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

Злотник

2015

Finite Difference Methods,Theory and Applications

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“…We consider the Schrödinger equation (2) in the simplest case V (x) = 0, scale time t so that a = c / = 1 and take the Gaussian wave packet…”

Section: Numerical Experimentsmentioning

confidence: 99%

A “converse” stability condition is necessary for a compact higher order scheme on non-uniform meshes for the time-dependent Schrödinger equation

Злотник

Čiegis

2018

Applied Mathematics Letters

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“…In our previous paper [5] we have investigated the performance of the standard Crank-Nicolson scheme supplemented with different BCs, including the exact discrete transparent boundary conditions (DTBCs) [8], and the approximate DTBCs suggested by Szeftel [26]. High order numerical approximations of BCs on nonuniform space grids are considered in [7,21].…”

Section: Mathematical Modelmentioning

confidence: 99%

Effective Numerical Algorithm for Simulations of Beam Stabilization in Broad Area Semiconductor Lasers and Amplifiers

Radziunas

Čiegis

2014

Mathematical Modelling and Analysis

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Abstract. A 2 + 1 dimensional PDE traveling wave model describing spatial-lateral dynamics of edge-emitting broad area semiconductor devices is considered. A numerical scheme based on a split-step Fourier method is presented. The domain decomposition method is used to parallelize the sequential algorithm. The parallel algorithm is implemented by using Message Passing Interface system, results of computational experiments are presented and the scalability of the algorithm is analyzed. Simulations of the model equations are used for optimizing of existing devices with respect to the emitted beam quality, as well as for creating and testing of novel device design concepts.

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On a family of finite-difference schemes with approximate transparent boundary conditions for a generalized 1D Schrödinger equation

Cited by 19 publications

References 11 publications

Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

Error Estimates of the Crank-Nicolson-Polylinear FEM with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped

A “converse” stability condition is necessary for a compact higher order scheme on non-uniform meshes for the time-dependent Schrödinger equation

Effective Numerical Algorithm for Simulations of Beam Stabilization in Broad Area Semiconductor Lasers and Amplifiers

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