Numerical study of the discontinuous Galerkin method for solving the Baer–Munziato equations with instantaneous mechanical relaxation

The work is devoted to a numerical study of the discontinuous Galerkin method for solving the two-phase Baer–Nunziato equations with instantaneous mechanical relaxation. From a mathematical point of view, the system of equations is a non-conservative hyperbolic system of equations. Unlike conservative hyperbolic systems of equations for which numerical methods are well known and developed, the numerical solution of non-conservative hyperbolic systems is a more complex problem that requires a generalization of the Godunov method. The computational algorithm is based on solving the hyperbolic part by a 2nd order discontinuous Galerkin method with path-conservative HLL or HLLEM numerical flows. To monotonize the solution, the WENO-S limiter is used, which is applied to the conservative variables of the model. To take into account relaxation processes, a new algorithm for instantaneous relaxation is proposed, within which the determination of equilibrium values of velocity and thermodynamic variables is reduced to solving a system of algebraic equations. To test the proposed numerical algorithm, the results of numerical calculations are compared with known analytical solutions in one-dimensional formulations. To demonstrate the capabilities of the proposed algorithms, a spatially two-dimensional problem of flow around a step is considered, as well as a two-phase version of the triple point problem. The calculation results show that the proposed algorithm is robust and allows calculations for two-phase media with a density jump of $\sim$1000.