Numerical implementation of traveling-wave solutions in heterogeneous media with two pressures taking into account the volume concentration of particles

The motion of shock waves through a mixture of gas and fine solid particles is studied using a model of the mechanics of heterogeneous media taking into account the difference in phase velocities, the intergranular pressure of the particles, and their finite volume concentration. The equation of state includes the volume concentration of particles. For this model, we have analytically and numerically studied the question of what types of strong discontinuities (frozen, dispersive, frozen-dispersive, two-front) exist and are stable in the dispersed medium considered and under what conditions they occur. A numerical solution of initial-boundary-value problems is obtained using a third-order TVD scheme for approximation in space and a five-order Runge-Kutta scheme for time approximations. The corresponding solutions in the class of traveling waves are calculated. A map of flow regimes in the form of shock waves is constructed, which makes it possible to determine the final velocity of the mixture behind the shock wave as a function of its initial velocity and the relative mass concentration of the gas phase. The above-mentioned stationary solutions for various types of shock waves are found numerically. Their stability is investigated by solving the Cauchy problem for nonstationary one-dimensional equations of the mechanics of heterogeneous media.