On the existence of non-negative numerical solutions of the Boltzmann equation

The diamond difference scheme approximating the linear Boltzmann equation may provide partly negative solutions. From the physics' point of view the solutions describing the density of neutrons or photons should be non-negative. It is shown that under assumptions being satisfied by suitable physical problems the solutions are non-negative if the step size is sufficiently small. This is shown for inhomogeneous boundary problems and for eigenvalue problems of the one-dimensional Boltzmann equation. In the latter case the greatest eigenvalue is a measure of the reactivity of reactors. It is proved that this eigenvalue is real and positive.