On Solving Singular Diffusion Equations With Monte Carlo Methods

Diffusion equations in one, two, or three dimensions with inhomogeneous diffusion coefficients are usually solved with finite-difference or finite-element methods. For higher dimensional problems, Monte Carlo solutions to the corresponding stochastic differential equations can be more effective. The inhomogeneities of the diffusion constants restrict the time steps. When the coefficient in front of the highest derivative of the corresponding differential equation goes to zero, the equation is said to be singular. For a 1-D stochastic differential equation, this corresponds to the diffusion coefficient that goes to zero, making the coefficient strongly inhomogeneous, which, however, is a natural condition when the process is limited to a region in phase space. The standard methods to solve stochastic differential equations near the boundaries are to reduce the time step and to use reflection. The strong inhomogeneity at the boundary will strongly limit the time steps. To allow for longer time steps for Monte Carlo codes, higher order methods have been developed with better convergence in phase space. The aim of our investigation is to find operators producing converged results for large time steps for higher dimensional problems. Here, we compare new and standard algorithms with known steady-state solutions.