Invariant imbedding, particle interaction, and conservation relations

Abstract

In a series of papers, we have applied invariant imbedding to provide new analytic and computational approaches to a variety of processes of mathematical physics. We began a detailed analysis of the ordinary and partial differential equations of invariant imbedding, concentrating upon existence and uniqueness of solution, nonnegativity of solution, and convergence of associated difference algorithms as step-size went to zero. In this paper, we wish to show that for a quite general class of transport processes involving particle-particle interaction as well as the usual particle-medium interaction, we can obtain difference approximations which exhibit nonnegativity and boundedness in an immediate fashion. Furthermore, a uniform Lipschitz condition is preserved. In subsequent papers, we shall discuss the more difficult matters of convergence to the solution of the partial differential equation and existence of this solution over the entire physical range of the independent variables.