Invariant Finite-Difference Schemes for Plane One-Dimensional MHD Flows That Preserve Conservation Laws

Invariant finite-difference schemes are considered for one-dimensional magnetohydrodynamics (MHD) equations in mass Lagrangian coordinates for the cases of finite and infinite conductivity. The construction of these schemes makes use of results of the group classification of MHD equations previously obtained by the authors. 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. New, previously unknown, conservation laws are found using symmetries and direct calculations. In the case of infinite conductivity, conservative invariant schemes are constructed as well. For isentropic flows of a polytropic gas the 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. In addition, 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.