Symbolic Computations in Simulations of Hydromagnetic Dynamo

Gleb Vodinchar,Lyubov Feshchenko,B Shevtsov,I Myagkova,V Kozlov

doi:10.1051/e3sconf/20172002017

Abstract

The compilation of spectral models of geophysical fluid dynamics and hydromagnetic dynamo involves the calculation of a large number of volume integrals from complex combinations of basis fields. In this paper we describe the automation of this computation with the help of systems of symbolic computations.

Highlights

Many problems of geophysical fluid dynamics reduce to the solution of the Navier-Stokes equations in a rotating coordinate system in an area having spherical symmetry [1]
In the most general case we can speak of a spherical shell
It should be noted that the construction of the system may require much greater computational effort than its subsequent solution. This is due to the fact that the calculation of the coefficients of the model reduces to the calculation of volume integrals over the spherical shell from various multiplicative combinations of the modes themselves and/or the operator ∇

Summary

Introduction

Many problems of geophysical fluid dynamics reduce to the solution of the Navier-Stokes equations in a rotating coordinate system in an area having spherical symmetry [1]. The spectral approaches are very popular, when solutions of the non-stationary problem are represented as a linear combination of a finite number of stationary basis fields (modes) with timedependent amplitudes For these amplitudes, using one of the weighted residuals methods, a dynamic system is composed. It should be noted that the construction of the system may require much greater computational effort than its subsequent solution This is due to the fact that the calculation of the coefficients of the model reduces to the calculation of volume integrals over the spherical shell from various multiplicative combinations of the modes themselves and/or the operator ∇. We consider the described range of problems and their solution in more detail

The original equations and basic systems

The equations of the model

The use of symbolic computations for the compilation of model equations

Conclusion