Topic/Type: 2.2 Kinetic methods, Particle-In-Cell and Vlasov, Poster

On Solving Singular Diffusion Equations with Monte-Carlo Methods

Q. Mukhtar , T. Hellsten

Alfv?n Lab, School of Electric Engineering, Royal Institute of Technology, Association VR-EURATOM, SE-100 44 Stockhom, Sweden

The classical method for solving diffusion equations is by making a convolution with the appropriate Green function. For constant diffusion coefficients the solution can readily be obtained. Diffusion equations in 1D or 2D with inhomogeneous diffusion coefficients are usually solved with finite difference or finite element methods. For higher dimensions Monte Carlo methods, stochastic differential equations, can become 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 1D stochastic differential equation this corresponds to that the diffusion coefficient goes to zero making the coefficient strongly inhomogeneous, which, however, is a natural condition when the process is limited to a region in the 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. Here we compare new and standard algorithms with known steady state solutions.