A Monte Carlo solution of transport equations

Abstract

A local method is developed for solving partial differential transport equations. The method is local in the sense that the value of the unknown solution of these equations can be calculated at arbitrary space and time coordinates directly rather than extracting its value from the field solution as done when using current numerical methods for solution. The proposed method is based on an analogy between the partial differential operator of transport equations and the infinitesimal generator of Itô processes, the Itô formula, the Dynkin formula, and Monte Carlo simulation. The method can be applied to solve transport problems with Dirichlet and Neumann boundary conditions. The solution of transport problems with Neumann boundary conditions is less simple because it requires the use of reflected Brownian motion and Itô processes.