Invariant finite-difference schemes with conservation laws preservation for one-dimensional MHD equations

TL;DR

This paper develops invariant finite-difference schemes for one-dimensional MHD equations that preserve conservation laws and symmetries, including new schemes for finite and infinite conductivity cases, validated through numerical testing.

Contribution

It introduces new invariant finite-difference schemes for MHD equations that exactly preserve symmetries and conservation laws, including previously unknown laws.

Findings

Schemes preserve all original symmetries and conservation laws.

Numerical tests confirm accurate conservation law preservation.

New schemes effectively handle finite and infinite conductivity cases.

Abstract

Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. For construction these schemes previously obtained results of the group classification of MHD equations are used. On the basis of the classical Samarskiy-Popov scheme new schemes are constructed for the case of finite conductivity. These schemes admit all symmetries of the original differential model and have difference analogues of all of its local differential conservation laws. Among the conservation laws there are previously unknown ones. In the case of infinite conductivity, conservative invariant schemes constructed as well. For isentropic flows of a polytropic gas proposed schemes possess the conservation law of energy and preserve entropy on two time layers. This is achieved by means of specially selected approximations for the equation of state of a polytropic gas. Also, invariant difference schemes with additional conservation laws are proposed. A new scheme for the case of finite conductivity is tested numerically for various boundary conditions which shows accurate preservation of difference conservation laws.