Systems of coupled Riccati equations in discrete kinetic theory: New self-similar solutions, shock profiles, stability

Abstract

Abstract In the kinetic theory of the discrete Boltzmann models (DBMs) the standard similarity shock waves solutions are, like the Broadwell shock waves, solutions of scalar Riccati equations. The corresponding microscopic densities, as well as the macroscopic conservative quantities: mass, momentum and energy, are monotonic. In the first part of the paper we construct new exact similarity shock waves solutions of systems of coupled Riccati equations, which arise for the DBMs with n conservation laws and n + k independent densities (here with n = 3, k = 2). The former standard shock waves are rational functions of two linear polynomials with an exponential variable while for the new waves the polynomials are quadratic and consequently nonmonotonic behaviours are possible. We stress that from the mathematical point of view we enter into a new domain of exact self-similar solutions associated to the systems of coupled Riccati equations. Due to the fact that, for the considered models, there exists a linear differential relation between two microscopic densities we obtain two different classes of solutions. For one class these two densities are necessarily of the former standard type while for the other all densities are of the new type of self-similar solutions. Furthermore we prove an important result: Let us consider microscopic densities associated to opposite velocities along the shock-axis. To any solution, there exists another one, called partner solution, such that those densities are exchanged with opposite similarity variable. This means that from any solution with mass, energy and pressure given, there exists a partner solution where these macroscopic quantities have opposite similarity variable. The new exact solutions are not the most general self-similar solutions of the considered models. In the second part we reduce the problem to the dynamical system of the coupled Riccati equations, with the limit conditions corresponding to the equilibrium states on both sides of the shocks. We analyse the structure of singularities corresponding to the equilibria, and obtain new nonmonotonic profiles of the microscopic densities by solving numerically the relevant two-point BVPs. For the new class of models studied in this paper we observe overshoots on one density for the exact solutions. In addition in the second part we find undershoots on one density and/or overshoot on one density accompanied by undershoots on another one. Contrary to the previously studied models, no nonmonotonic effects have been observed for the macroscopic conserved quantities: mass and energy.