Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection-diffusion-reaction problems

In this paper a scheme for approximating solutions of convection-diffusion-reaction equations by Markov jump processes is studied. The general principle of the method of lines reduces evolution partial differential equations to semi-discrete approximations consisting of systems of ordinary differential equations. Our approach is to use for this resulting system a stochastic scheme which is essentially a direct simulation of the corresponding infinitesimal dynamics. This implies automatically the time adaptivity and, in one space dimension, stable approximations of diffusion operators on non-uniform grids and the possibility of using moving cells for the transport part, all within the framework of an explicit method. We present several results in one space dimension including free boundary problems, but the general algorithm is simple, flexible and on uniform grids it can be formulated for general evolution partial differential equations in arbitrary space dimensions.