A finite-difference method for the one-dimensional time-dependent schrödinger equation on unbounded domain

Solutions for One-Dimensional Time-Dependent Schrödinger Equations

“…In a meshless (meshfree) method, a set of scattered nodes are selected in the computational domain. Meshless schemes include the method of fundamental (MFS) [2], the method of particular solutions (MPS) [3], the element-free Galerkin method [4], and local point interpolation [5], boundary knot method, etc. To extend the MFS to inhomogeneous equations or timedependent problems, the MPS has been introduced to evaluate the particular solution of the given differential equation.Since the particular solution is not unique, there is a rich variety Radial basis functions(RBFs), polynomial functions, trigonometric functions etc [6,7,8,9,10], have been employed as the basis functions to approximate the particular solutions for the given differential equation.…”

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

“…Using the approach in [13] to discretize the boundary conditions (1.9) and (1.10), we have (see [13])…”

“…Unfortunately, the resulting schemes have been proved to be conditionally stable and the strong numerical reflections can be induced. Recently, Wu et al developed a new, modified BP approach for the boundary conditions of the heat equation [26] and the Schrödinger equation [13], the discretized boundary conditions are exact in spatial direction, the resulting finite difference scheme is unconditionally stable, and almost no numerical reflections appear at the boundaries.…”

“…The approaches proposed in these papers are efficient, and the numerical examples also show that the numerical schemes have good accuracy, but no rigorous global error estimates are given for the fully discrete solutions. In this paper, we apply the approach proposed in [13] to discretize the boundary conditions (1.9) and (1.10), the Crank-Nicolson scheme in time and linear or quadratic finite element approximation in space to discretize Eq. (1.8).…”

Spectral Collocation Technique for Absorbing Boundary Conditions with Increasingly High Order Approximation

“…There are many papers [1,2,3,4,5,6,7,8] developing TBCs and studying their difference approximations and stability. However, the obtained TBC through this way is nonlocal in t, thus requiring all the history data in memory.…”