We study the shock wave problem for the general discrete velocity model
(DVM), with an arbitrary finite number of velocities. In this case the
discrete Boltzmann equation becomes a system of ordinary differential
equations (dynamical system). Then the shock waves can be seen as
heteroclinic orbits connecting two singular points (Maxwellians). In this
paper we give a constructive proof for the existence of solutions in the
case of weak shocks.
¶ We assume that a given Maxwellian is approached at infinity, and consider
shock speeds close to a typical speed, corresponding to the sound speed in
the continuous case. The existence of a non-negative locally unique (up to a
shift in the independent variable) bounded solution is proved by using
contraction mapping arguments (after a suitable decomposition of the
system). This solution is shown to tend to a Maxwellian at minus infinity.
¶ Existence of weak shock wave solutions for DVMs was proved by Bose, Illner
and Ukai in 1998. In this paper, we give a constructive, more
straightforward, proof that suits the discrete case. Our approach is based
on earlier results by the authors on the main characteristics (dimensions of
corre- sponding stable, unstable and center manifolds) for singular points
of general dynamical systems of the same type as in the shock wave problem
for DVMs.
¶ The same approach can also be applied for DVMs for mixtures.