Positive (2+1)-dimensional exact shock waves solutions to the Broadwell model

Abstract

The exact solutions for the discrete Boltzmann models are sums of similarity shock waves: at least three for the (2+1)-dimensional solutions (two spatial coordinates). Here, for the three-dimensional, six velocities, Broadwell model shock waves solutions are also constructed. The difficult problem is the determination of solutions with positive density. It is proven that in the arbitrary parameters space, from which the solutions are constructed, there exists a domain leading to positivity. As an illustration, a numerical example is discussed. In the spatial coordinate plane both shock waves along one axis and bumps along the other are observed. The movement of these structures with time is discussed.