Splitting in Potential Finite-Difference Schemes with Discrete Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation

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

doi:10.1007/978-3-319-10705-9_20

Cited by 2 publications

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References 9 publications

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“…For n = 1, the results are as well particular cases of those from [13] (given specifically for general FEM). The case n ≥ 3 can be treated in the same manner as n = 2 (for such an example, see [11]).…”

Section: The Ibvp and Numerical Methods To Solve Itmentioning

confidence: 99%

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: 99%

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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On a Numerov–Crank–Nicolson–Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

Злотник

Romanova

2015

Applied Numerical Mathematics

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We consider an initial-boundary value problem for a 2D time-dependent Schrödinger equation on a semiinfinite strip. For the Numerov-Crank-Nicolson finite-difference scheme with discrete transparent boundary conditions, the Strang-type splitting with respect to the potential is applied. For the resulting method, the uniqueness of a solution and the uniform in time L 2 -stability (in particular, L 2 -conservativeness) are proved. Due to the splitting, an effective direct algorithm using FFT in the direction perpendicular to the strip is developed to implement the splitting method for general potential. Numerical results on the tunnel effect for smooth and rectangular barriers together with the practical error analysis on refining meshes are included as well.Keywords: the time-dependent Schrödinger equation, the Numerov discretization in space, the Crank-Nicolson discretization in time, the Strang splitting, discrete transparent boundary conditions, uniqueness, stability, tunnel effect, practical error analysis MSC[2010] 65M06, 65M12, 35Q41. IntroductionThe time-dependent Schrödinger equation with several variables is important in quantum mechanics, atomic and nuclear physics, wave physics, microelectronics and nanotechnologies, etc. Often it should be solved in unbounded space domains.Several approaches were developed and studied to deal with problems of such kind, in particular, see [1,2,3,6,18,20]. One of them exploits the so-called discrete transparent boundary conditions (TBCs) at artificial boundaries [3,11]. Its advantages are the complete absence of spurious reflections, reliable computational stability, clear mathematical background and corresponding rigorous stability theory.Concerning finite-difference schemes in several space variables with the discrete TBCs, the standard Crank-Nicolson scheme in the case of an infinite (or semi-infinite) strip was studied in detail in [3,7,8]; the higher order Numerov-Crank-Nicolson scheme was considered in [19], and a general family of schemes was treated in [23,24]. But all the schemes are implicit, and solving of specific complex systems of linear algebraic equations is required at each time level. Efficient methods to solve similar systems are well developed by the moment in real situation but not the complex one.The splitting technique is widely used to simplify numerical solving of the time-dependent Schrödinger and related equations, in particular, see [4,5], [13]-[17], [21]. The Strang-type splitting with respect to the *

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Splitting in Potential Finite-Difference Schemes with Discrete Transparent Boundary Conditions for the Time-Dependent Schrödinger Equation

Cited by 2 publications

References 9 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

On a Numerov–Crank–Nicolson–Strang scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip

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