Difference Scheme with a Symmetry-Analyzer for Equations of Gas Dynamics and Magnetohydrodynamics

The chapter describes an approach using Godunov-type difference schemes to construct numerical algorithms for numerically integrating equations of ideal gas dynamics and magnetohydrodynamics. This approach allows calculating the flows of conservative variables through the interface between the computational cells. The Riemann problem is solved (exactly or approximately) on the decay of the discontinuity between the states that are formed using some procedures for reconstructing grid data to this interface. During reconstructing vector quantities, the question arises: what components of vector fields should be preferred during the reconstruction? To automate this choice, we propose a symmetry-analyzer as an element of the computational algorithm for the numerical solution of two-dimensional equations of ideal gas dynamics and magnetohydrodynamics. A symmetry-analyzer is an algorithm that allows using grid data to give a preference to a component (in the present work, Cartesian or polar) of a vector field for its reconstruction on the cell interfaces of a computational grid. A computational algorithm is constructed using a polar-type computational grid with a symmetry-analyzer. The algorithm is easily transferred to a three-dimensional cylindrical type of computational grids.