Numerical treatment of a nonlinear nonlocal transport equation modeling crystal precipitation

Numerical methods to solve certain nonlinear nonlocal transport equations (hyperbolic partial differential equations with smooth solutions), even singular at the boundary, are developed and analyzed. As a typical case, a model equation used to describe certain crystal precipitation phenomena (a slight variant of the so-called Lifshitz-Slyozov-Wagner model) is considered. Choosing a train of few delta functions as initial crystal size distribution, one can model the technologically important case of having only a modest number of crystal sizes. This leads to the reduction of the transport equation to a system of ordinary differential equations, and suggests a new method of solution for the transport equation, based on Shannon sampling, which is widely used in communication theory.