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

Abstract

An initial-boundary value problem for the n-dimensional (n 2) time-dependent Schrödinger equation in a semi-infinite (or infinite) parallelepiped is considered. Starting from the Numerov-Crank-Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for n 3. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting method, the uniqueness of solution and the unconditional uniform in time L 2 -stability (in particular, L 2 -conservativeness) are proved. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applicable for general potential. Numerical results on the 2D tunnel effect for a Pöschl-Teller-like potential-barrier and a rectangular potential-well are also included.MSC[2010] classification: 65M06, 65M12, 35Q40. Keywords: the time-dependent Schrödinger equation, the Crank-Nicolson finite-difference scheme, higher-order scheme, the Strang splitting, discrete transparent boundary conditions, stability, tunnel effect IntroductionThe time-dependent Schrödinger equation with several space variables is crucial in quantum mechanics and electronics, nuclear and atomic physics, wave physics, etc. Often it should be solved in unbounded space domains.Several approaches were developed and investigated for solving problems of such kind, in particular, see [1,2,3,6,16,18]. 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 in practice as well as the rigorous mathematical background and stability results in theory.The discrete TBCs for the Crank-Nicolson finite-difference scheme, the higher order Numerov-Crank-Nicolson scheme and a general family of schemes on an infinite or semiinfinite strip were constructed and studied respectively in [3,7,8], [17] and [21,22]. 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...

On Compact 4th Order Finite-Difference Schemes for the Wave Equation

“…But the latter scheme fails for n ≥ 3 similarly to [3] in the case of the TDSE. The point is that s N should approximate I adequately, but for the minimal and maximal eigenvalues of s N < I as the operator in H h we have…”

“…Its corollary rigorously ensures the 4th order error bound in the case of smooth solutions to the IBVP. Note that stability is unconditional for similar compact schemes on uniform meshes for other type PDEs, for example, see [3,11]. Our approach is applied in a unified manner for any n ≥ 1 (not separately for n = 1, 2 or 3 as in many papers), the uniform rectangular (not only square) mesh is taken, the stability results are of standard kind in the theory of finite-difference schemes and proved by the energy techniques (not only by getting bounds for harmonics of the numerical solution as in most papers).…”

Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations

“…The limit of vanishing spatial approximation parameters in the DTBC coincides with the temporally semi-discrete transparent boundary conditions of [22,19]. DTBC for the Schrödinger equation were also derived for finite-element [30] and splitting higher-order schemes [14].…”

“…x } (see (12)). The spatial grid X PML is the same as in the stationary case (see (14)). Using the Crank-Nicolson time-integration method gives (22) but with the modified spatial differential operatorD 2…”

“…where X PML j 1 denotes the j 1 -th grid point of the one-dimensional grid X PML defined in (14). Without taking into account a possible incoming plane wave, we can state the problem in the following form:…”