A modification of the large-particle method to a scheme having the second order of accuracy in space and time for shockwave flows in a gas suspension

We develop the previously proposed approach of constructing difference schemes for solving stiff problems of shockwave flows of heterogeneous media using an implicit non-iterative algorithm for calculating interactions between the phases. The large particle method is modified to a scheme having the second order of accuracy in time and space on smooth solutions. At the first stage, we use the central differences with artificial viscosity of TVD type. At the second stage, we implement TVD-reconstruction by weighted linear combination of upwind and central approximations with flow limiters. The scheme is supplemented by a two-step Runge-Kutta method in time. The scheme is K-stable, i.e. the time step does not depend on the intensity of interactions between the phases, but is determined by the Courant number for a homogeneous system of equations (without source terms). We use test problems to confirm the monotonicity, low dissipation, high stability of the scheme and convergence of numerical results to the exact self-similar equilibrium solutions in a gas suspension. Also, we show the scheme capability for numerical simulation of physical instability and turbulence. The method can be used for flows of gas suspensions having complex structure.