A particle method for some parabolic equations

We present a quasi-Monte-Carlo particle simulation of some multidimensional linear parabolic equations with constant coefficients. We approximate the elliptic operator in space by a finite-difference operator. We discretize time into intervals of length Δt. The discrete representation of the solution at time tn = nΔt is a sum of Dirac delta measures. Using the explicit Euler scheme, the resulting approximation at time tn+1 is recovered by a quasi-Monte-Carlo integration. We make use of a technique involving renumbering the simulated particles in every time step. We state and prove a convergence theorem for the method. Experimental results are presented for some model problems. The results suggest that the quasi-Monte-Carlo simulation tends to give more accurate solutions than a Monte-Carlo simulation, when the correct renumbering technique is used. Other choices can result in significant loss of efficiency.