Riccati-coupled similarity shock wave solutions for multispeed discrete Boltzmann models

Abstract

We study nonstandard shock wave similarity solutions for three multispeed discrete Boltzmann models: (1) the square 8Ν_i, model with speeds 1 and √2 with thex axis along one median, (2) the Cabannes cubic 14Ν_i model with speeds 1 and √3 and thex axis perpendicular to one face, and (3) another 14Ν_i, model with speeds 1 and √2. These models have five independent densities and two nonlinear Riccati-coupled equations. The standard similarity shock waves, solutions of scalar Riccati equations, are monotonic and the same behavior holds for the conservative macroscopic quantities. First, we determine exact similarity shock-wave solutions of coupled Riccati equations and we observe nonmonotonic behavior for one density and a smaller effect for one conservative macroscopic quantity when we allow a violation of the microreversibility. Second, we obtain new results on the Whitham weak shock wave propagation. Third, we solve numerically the corresponding dynamical system, with microreversibility satisfied or not, and we also observe the analogous nonmonotonic behavior.