SIMULATION OF DIFFUSION USING QUASI-RANDOM WALK METHODS

We present a method for computer simulation of diffusion. The method u ses quasi-random walk of particles. We consider a pure initial value p roblem for a simple diffusion equation in s space dimensions. We intro duce s spatial steps Delta x(i). A semidiscrete approximation to the e quation is obtained by replacing the spatial derivatives with finite d ifferences. N particles are sampled from the initial distribution. The time interval is partitioned into subintervals of length Delta t. The discretization in time is obtained by resorting to the forward Euler method. In every time step the particle movement is regarded as an app roximate integration is s+1 dimensions. A quasi-Monte Carlo estimate f or the integral is obtained by using a (0, s+1)-sequence. A key elemen t in successfully applying the low discrepancy sequence is a technique involving renumbering the particles at each time step. We prove that the computed solution converges to the solution of the semi-discrete e quation as N-->infinity and Delta t-->0. We present numerical tests wh ich show that random walk results are improved with quasi random seque nces and renumbering. (C) 1998 IMACS/Elsevier Science B.V.