Exact solutions of the Broadwell model in 1+1 dimensions

Abstract

The author studies the one spatial dimensional, 6-velocity Broadwell model with four identical densities and three independent ones. The author determines 'solitons' (one-dimensional shock wave solutions) and 'bisolitons' (two-dimensional, space plus time solutions) which are rational fractions with one or two exponential variables. The author obtains three classes of positive exact solutions in 1+1 dimensions (space x, time t). The first one is periodic in the space variable and for large time the solutions correspond to propagating damped linear waves. The second is positive only along one semi x axis while the third, positive along the whole x axis, represents non-planar damped shock waves. Using the same tools in a companion paper, for the discrete 2-velocity models, the author obtains in a two-dimensional space the first two classes of solutions mentioned above. This suggests that, for the discrete Boltzmann models, general methods exist for the determination of non-trivial exact solutions.