We describe an algorithm for animating time-dependent quantum wave functions in one dimension with very high accuracy. The algorithm employs the Crank–Nicholson approximation for the time dependence along with a Numerov extension of the discrete transparent boundary conditions described recently by Ehrhardt. We illustrate the power of this approach by simulating the decay of alpha particles fro...

Abstract. Second-order boundary value problems are solved with exponentially-fltted Numerov methods. In order to attribute a value to the free parameter in such a method, we look at the leading term of the local truncation error. By solving the problem in two phases, a value for this parameter can be found such that the tuned method behaves like a sixth order method. Furthermore, guidelines to ...

We study necessary conditions for stability of a Numerov-type compact higher-order finite-difference scheme the 1D homogeneous wave equation in case non-uniform spatial meshes. first show that uniform time cannot be valid any norm provided complex eigenvalues appear associated mesh eigenvalue problem. Moreover, we prove then solution grows exponentially making strongly non-dissipative and there...

In this paper, a class of third order singularly perturbed boundary value problems with suitable boundary conditions is considered. The third order boundary value problem is transformed to asymptotically equivalent second order boundary value problem. This problem is solved efficiently by using fitted Numerov method. Linear and non-linear examples are solved to illustrate the method and relativ...

Numerov's numerical method is developed in a didactic way by using Python its {\it Jupyter Notebook} version 6.0.3 for three different quantum physical systems: the hydrogen atom, molecule governed Morse potential and dot. After brief introduction to Numerov method, complete code calculate eigenfunctions eigenvalues of atom presented. The necessary changes other two examples are also provided s...

Abstract A functionally-fitted Numerov-type method is developed for the numerical solution of second-order initial-value problems with oscillatory solutions. The basis functions are considered among trigonometric and hyperbolic ones. characteristics studied, particularly, it shown that has a third order convergence general ordinary differential equation, $$y''=f \left( x,y,y' \right) $$ <mml:ma...

Abstract The Numerov process is a solution method applicable to some classes of differential equations, that provides an error term the fifth order in grid size with computational cost comparable finite-difference scheme. In original formulation method, uniform required; paper shows procedure for extending its applicability non-uniform one dimension. effectiveness tested on model problem, and c...