Exact solutions for nonconservative discrete velocity models

Abstract

Extended discrete kinetic theory including sources, sinks, creation and absorption of test particles, inelastic scattering,… added to the elastic collisions, was introduced by Boffi and Spiga. For the mass conservation law (or momentum, energy), polynomials of the mass (or densities) are added, leading to a lack of conservation equations. We consider models with linear nonconservative terms, LNC (Spiga- Platkowski) and quadratic, QNC (Piechor-Platkowski). For quasi-linear systems of PDE (linear differential terms and quadratic nonlinearities), we present a general formalism for the determination of stationary, similarity, periodic and (1 + 1) dimensional exact solutions in one space variable. We present results for the two-velocity, the two and three dimensional Broadwell, the hexagonal 6vi and the two-squares 8vi DVMs (Discrete Velocity Models). The similarity solutions are obtained from the compatibility between different scalar nonlinear Riccati equations (NLODE), Firstly, we apply this method to the similarity solutions of the QNC models, while for (1 + 1) solutions only one density is not constant. Secondly for similarity solutions of the LNC models, we require that, like in conservative models, the jump relations (mass, momentum.) be satisfied and obtain restrictions on both the nonconservative and solutions parameters. Thirdly, for QNC (also LNC for 2vi ) we restrict only the nonconservative parameters such that the mass conservation is restored. The similarity solutions can be interpreted as shock waves and we check both the Whitham-Lax criteria and the shock inequalities. Like for conservative models, we find (1 + 1) solutions which are sums of similarity waves.