We construct exact (1 + l)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x/> 0, t t> 0.
For the discrete Boltzmann models, the velocities can only take discrete values v,., I v, I = 1, i = 1, ..., 2p with p couples of opposite velocities v2j_ 1, v2j, J = 1 . . . . . p. To each v i is associated a density N~(x, t) (x space variables and time t). It has recently been understood how to construct (1 + 1)-dimensional [1] (space x and time t) and (2 + 1)dimensional [2] (space x, y and time t) exact solutions. They simply are the sums of two (one spatial dimension) or three (two spatial dimensions) similarity shock waves. Except for a few cases (periodic solutions for the Carleman model [3] or Broadwell model with or without ternary collisions [1]) these exact solutions, obtained without boundary conditions, cover the whole available space (x-axis or x, y plane).
Although it is physically interesting to consider a semi-infinite medium, with a pure specular reflection boundary condition, due to additional constraints, in general it is more difficult to tackle the exact solutions. However, we have been able to construct positive (1 + 1)-dimensional exact solutions for particles in the presence of a purely reflecting wall. Here we present the simplest result.
Let us consider the square velocity discrete model [4], attributed to Maxwell, with VI, u u d-u = 0 along the x-axis and u u u + u = 0 along the y-axis. For an elastic wall at x = 0, the specular reflection boundary condition is