A high order method for the solution of the single channel Schrödinger equation

1985 ◽  
Vol 54 (3) ◽  
pp. 733-740 ◽  
Author(s):  
Paul L. Devries
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Single Channel ◽  
High Order ◽  
Order Method ◽  
High Order Method

The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation

2015 ◽  
Vol 15 (2) ◽  
pp. 233-245 ◽  
Author(s):  
Alexander Zlotnik ◽  
Ilya Zlotnik
Keyword(s):  
Cauchy Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
High Order ◽  
Standard Theory ◽  
Order Method ◽  
Transparent Boundary Conditions ◽  
Time Step ◽  
High Order Method ◽  
The Cauchy Problem

AbstractWe consider the Cauchy problem for the 1D generalized Schrödinger equation on the whole axis. To solve it, any order finite element in space and the Crank–Nicolson in time method with the discrete transparent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order method both with respect to space and time steps. To study its properties, we comment on its stability and give results of numerical experiments and enlarged practical error analysis for three typical examples. Unlike most common second order (in either space or time step) methods, the proposed method is able to provide high precision results in the uniform norm by using adequate computational costs. It works even in the case of discontinuous potentials and non-smooth solutions far beyond the scope of its standard theory. Comparing our results to the previous ones, we obtain much more accurate results using much less amount of both elements and time steps.

scholarly journals Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 761
Author(s):  
Călin-Ioan Gheorghiu
Keyword(s):  
Schrödinger Equation ◽  
Small Parameter ◽  
Error Estimation ◽  
Schrodinger Equation ◽  
Error Control ◽  
High Order ◽  
Order Of Approximation ◽  
Schrödinger Equations ◽  
Spectral Collocation ◽  
Integration Interval

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

Theoretical Aspects of a Multidomain High-Order Method for CAA

2003 ◽  
Author(s):  
Ronan Guenanff ◽  
Pierre Sagaut ◽  
Eric Manoha ◽  
Marc Terracol ◽  
Roger Lewandowsky
Keyword(s):  
High Order ◽  
Order Method ◽  
High Order Method

Analysis of a filtered high-order method for CAA

2003 ◽  
pp. 1978-1981 ◽  
Author(s):  
R. Guénanff ◽  
P. Sagaut ◽  
E. Manoha ◽  
R. Lewandowski
Keyword(s):  
High Order ◽  
Order Method ◽  
High Order Method
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