Numerical Study of the Discontinuous Galerkin Method for Solving the Baer–Nunziato Equations with Instantaneous Mechanical Relaxation

This paper is a numerical study of the discontinuous Galerkin method for solving the two-phase Baer–Nunziato (BN) equations with instantaneous mechanical relaxation. From a mathematical point of view, the system of equations is a nonconservative hyperbolic system of equations. Unlike conservative hyperbolic systems of equations for which numerical methods are well known and developed, the numerical solution of nonconservative hyperbolic systems is a more complex problem that requires generalization of the Godunov method. The computational algorithm is based on solving the hyperbolic part by a second 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 the 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 the 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 ~1000.