Effective Numerical Approximation of Schrödinger type Equations through Multiderivative Exponentially‐fitted Schemes

Abstract

In this paper an exponentially-fitted multiderivative method is developed for the numerical integration of the Schrödinger equation. We call the method multi-derivative since it uses derivatives of orders two and four. An application to the resonance problem of the radial Schrödinger equation indicates that the new method is more efficient than the Numerov method and other known methods found in the literature.

Numerical multistep methods for the efficient solution of quantum mechanics and related problems

“…In [32] Thomas, Simos and Mitsou have developed a family of Numerov-type exponentially fitted hybrid methods. In [33] Psihoyios and Simos have constructed an implicit hybrid method. In [34] Simos has developed a trigonometrically fitted hybrid method.…”

A New Symmetric p-stable Obrechkoff Method with Optimal Phase lag for Oscillatory Problems

“…Popular examples include: mechanical systems without dissipation, satellite tracking, celestial mechanics, etc. The solution of the type (1) which is considered in this paper is a priori known to be periodic, and when integrated numerically, the desire is that the numerical solution also preserves the analogical periodicity of the analytic solution [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Furthermore, equation ( 1) is known to have inherent "periodic stiffness" [12] which makes it difficult to solve analytically.…”

High phase-lag order trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems

“…For several decades, there has been strong interest in searching for better numerical methods to integrate first-order and second-order initial value problems, because these problems are usually encountered in celestial mechanics, quantum mechanical scattering theory, theoretical physics and chemistry, and electronics. Generally, the solution of (1) is periodic, so it is expected that the result produced by some numerical methods preserves the analogical periodicity of the analytic solution [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. Computational methods involving a parameter proposed by Gautschi [8], Jain et al [10], Sommeijer and et al [30] and Steifel and Bettis [31] yield numerical solution of problems of class (1).…”