A Difference Scheme with Well-Controlled Dissipation for Solving the Kapila Model Equations

This paper is devoted to the derivation and numerical studies of a difference scheme withwell-controlled dissipation for solving the Kapila model equations. The Kapila model is widelyused for analysis of two-phase compressible flows. It has a form of a first-order nonconservativehyperbolic system. As any other first-order nonconservative hyperbolic system, it requiresdetermining the regularizing dissipative operator to define discontinuous solutions andRankine–Hugoniot conditions. The choice of dissipative operator influences the wave structureobserved in the solutions. Schemes with well-controlled dissipation are constructed in such a waythat the dissipative operator determined by the form of their equivalent equation coincides withthe one used to define well-posed setting of the original problem to be solved. As a result, it isexpected that the numerical solution converges to the solution of the system under consideration.Numerical experiments presented in the paper demonstrate the efficiency of this approach.Numerical solutions of the traveling wave type obtained by other methods were used as exactsolutions.