On Comparison of Two Reliable Techniques for the Riesz Fractional Complex Ginzburg- Landau-Schr\"odinger Equation in Modelling Superconductivity

TL;DR

This paper compares two numerical techniques, an implicit finite difference method and a spectral method, for solving the Riesz fractional complex Ginzburg-Landau-Schrödinger equation, analyzing their stability and accuracy in modeling superconductivity.

Contribution

It provides a comparative analysis of IFDM and TSFS techniques for the Riesz fractional CGLS equation, including stability and error assessments.

Findings

TSFS is unconditionally stable.

Both methods' errors are tabulated for various fractional orders.

Spectral method shows higher accuracy in solutions.

Abstract

In the present paper, the Complex Ginzburg-Landau-Schr\"odinger (CGLS) equation with the Riesz fractional derivative has been treated by a reliable implicit finite difference method (IFDM) of second order and furthermore for the purpose of a comparative study, and also for the investigation of the accuracy of the resulting solutions another effective spectral technique viz. time-splitting Fourier spectral (TSFS) technique has been utilized. In the case of the finite difference discretization, the Riesz fractional derivative is approximated by the fractional centered difference approach. Further the stability of the proposed methods has been analysed thoroughly and the TSFS technique is proved to be unconditionally stable. Moreover the absolute errors for the solutions of |\chi(x, t)|^2 obtained from both the techniques for various fractional order have been tabulated. Further the L^2 and L^infinity error norms has been displayed for |\chi(x, t)|^2 and the results are also graphically depicted.