Two Optimized Symmetric Eight-Step Implicit Methods for Initial-Value Problems with Oscillating Solutions

TL;DR

This paper introduces two optimized symmetric eight-step implicit methods with phase-lag order ten and infinite phase-fitted order, designed for efficiently solving oscillating initial-value problems like the Schrödinger equation.

Contribution

The paper presents novel phase-fitted eight-step implicit methods with infinite phase-lag order, improving numerical efficiency for oscillatory problems compared to existing methods.

Findings

The infinite phase-lag method outperforms others in efficiency.

The methods are effective for Schrödinger equation and orbital problems.

Optimized methods reduce computational effort for oscillatory IVPs.

Abstract

In this paper we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr\"odinger equation with the use of the Woods-Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved.