Instability and stability of numerical approximations to discrete velocity models of the Boltzmann equations
We study a standard, explicit finite difference approximation of the 2-D Broadwell model and construct a numerical solution with the sum-norm growing in time faster than any polynomial. Our construction is based on a structure of a self-similar fractal! We also obtain global existence, long-time behavior and numerical stability results of a large class of multidimensional discrete velocity models of the Boltzmann equation. We assume certain restrictions on the size of the support and the sup-norm of the initial data. Our results are obtained by examining the time evolution of sets on which the solutions are supported.