Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme

Abstract

Abstract. We investigate the solvability of the Dirichlet boundary value problemwhere λ is a nonnegative parameter. We discuss the existence of multiple positive solutions and show that for certain values of λ, there also exist solutions that vanish on a subinterval [ρ, 1 − ρ] ⊂ (0, 1), the so-called dead core solutions. In order to illustrate the theoretical findings, we present computational results for g(u) = 1/ √ u, computed using the collocation method implemented in bvpsuite, a new version of the standard MATLAB code sbvp1.0.

A fast solution method for time dependent multidimensional Schrödinger equations

“…In all the figures we used the approximation formula of order N = 6, the extension of w j = w j and the step h = 0.005. Similar tests with finite difference scheme can be found in [1] and [14].…”

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

“…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].…”

Remarks on discrete and semi-discrete transparent boundary conditions for solving the time-dependent Schrödinger equation on the half-axis

“…For different values of θ, the family includes a number of particular schemes: the standard Crank-Nicolson scheme without averages (for θ = 0) studied in [3,9,6,7], the finite element method (FEM) for linear elements (for θ = 1 6 ) studied in particular in [2,14], a four-point symmetric vector (or multi-symplectic) scheme (for θ = 1 4 ) studied in equivalent forms in [10,11] and, in the case of constant coefficients (for θ = 1 12 ), the higher-order Numerov scheme presented in [12,17] (see also the 2D case in [15]). The case θ = 1 4 corresponds also to the linear FEM with the numerical integration based on the midpoint rule (in the integrals containing ρ and V ).…”

“…Several approaches were developed and investigated for solving problems of such kind in 1D, see review [1]. Among them one exploits the so-called discrete (both in space and time) transparent boundary conditions (DTBCs) at artificial boundaries, see [3,9,4,12,15] and [6,7,8,18]. Their advantages are the complete absence of spurious reflections in practice as well as the rigorous mathematical background and relevant stability results in theory.…”