Algorithm of the Parallel Sweep Method for Numerical Solution of the Gross–Pitaevskii Equation with Highest Nonlinearities

А Д Булыгин

doi:10.14529/jsfi180415

Abstract

In this paper, we for the first time introduce a numerical scheme the solution of a nonlinear equation of the Gross–Pitaevskii type (GP) or the nonlinear Schrodinger equation (NLSE) with highest nonlinearities, which provides implementation of a complete set of motion integrals. This scheme was parallelly implemented on a non-uniform grid. Propagation of a ring laser beam with non-zero angular momentum in the filamentation mode is studied using the implemented numerical scheme. It is shown, that filaments under exposure to centrifugal forces escape to the periphery. Based on a number of numerical experiments, we have found the universal property of motion integrals in the non-conservative case for a given class of equations. Research of dynamics of angular momentum for a dissipative case are also presented. We found, that angular moment, particularly normed by initial energy during filamentation process, is quasi-constant.

Highlights

A nonlinear parabolic partial differential equation occurs in many applications [3]
In [2], a wide range of numerical schemes used for solution of the Gross–Pitaevskii type (GP) (NLSE) equation with highest nonlinearities was constructed by the example of a case with radial symmetry, and it was determined that the simplest and quite effective numerical method is the method of splitting by physical factors method
Discrete difference methods for solving the NLS equation are optimal for tracking and suppressing numerical imbalances, and the adaptive step along the evolutionary coordinate should be selected according to the conditions of preserving the Hamilton function on the numerical solution of the GP (NLSE) equation

Summary

Introduction

A nonlinear parabolic partial differential equation (or a system of such equations) occurs in many applications [3]. Their rigorous analytical solutions are often unknown Such equations are solved by numerical methods. Discrete difference methods for solving the NLS equation are optimal for tracking and suppressing numerical imbalances, and the adaptive step along the evolutionary coordinate should be selected according to the conditions of preserving the Hamilton function on the numerical solution of the GP (NLSE) equation. This step is significantly less compared to those offered in other works. Implementation of the given exact relations in the numerical solution, even in the conservative case, imposes very strict conditions on the numerical grid, which makes the use of parallel algorithms to be relevant

The Numerical Scheme

Results of Numerical Calculations

Conclusion