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

Abstract

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 *

On a splitting higher-order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped

“…The splitting technique is widely used to simplify numerical solving of the timedependent Schrödinger and related equations, in particular, see [4,5,13,14,15,19]. The known Strang-type splitting with respect to the potential has been recently applied to the Crank-Nicolson and the Numerov-Crank-Nicolson scheme with the discrete TBCs in 2D case in [10,20].…”

“…Owing to the Strang-type splitting, an effective direct algorithm is considered to implement the method (for general potential) similar to those constructed in [10,20]. It uses the fast Fourier transform (FFT) in the perpendicular directions and a collection of independent 1D discrete Schrödinger problems at each time level.…”

“…It uses the fast Fourier transform (FFT) in the perpendicular directions and a collection of independent 1D discrete Schrödinger problems at each time level.The corresponding 2D numerical results on the tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are included. In the case of the potentialbarrier, we compare the "double-splitting" scheme with the Numerov-Crank-Nicolson-Strang scheme from [20] and find that their errors are very close. In the case the rectangular well, we present the behavior of the solution.…”

“…All these schemes are implicit, so to implement them, solving of specific complex systems of linear algebraic equations is required at each time level.The splitting technique is widely used to simplify numerical solving of the timedependent Schrödinger and related equations, in particular, see [4,5,13,14,15,19]. The known Strang-type splitting with respect to the potential has been recently applied to the Crank-Nicolson and the Numerov-Crank-Nicolson scheme with the discrete TBCs in 2D case in [10,20].Higher order methods are important due to their ability to reduce computational costs essentially, and the Numerov-Crank-Nicolson scheme can be written in n-dimensional case as well. But we show that, for n 3, the Numerov space operators lose their important spectral properties existing for n = 2 so that the scheme becomes impractical.In this paper, in the spirit of [21,22], we first split these operators (in space) and recover the properties without reducing the higher order, for any n 2.…”

“…Exploiting an approach from [20] and suitable results from [9], we derive the uniqueness of solution to the "double-splitting" scheme with the discrete TBC and then its uniform in time L 2 -stability (from the former stability result for the infinite space mesh). In particular, it is L 2 -conservative.Owing to the Strang-type splitting, an effective direct algorithm is considered to implement the method (for general potential) similar to those constructed in [10,20]. It uses the fast Fourier transform (FFT) in the perpendicular directions and a collection of independent 1D discrete Schrödinger problems at each time level.The corresponding 2D numerical results on the tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are included.…”