Convergence of a random particle method to solutions of the Kolmogorov equation 𝑢_{𝑡}=𝜈𝑢ₓₓ+𝑢(1-𝑢)

Abstract

We study a random particle method for solving the reaction-diffusion equation u t = ν u x x + f ( u ) {u_t} = u {u_{xx}} + f(u) which is a one-dimensional analogue of the random vortex method. It is a fractional step method in which u t = ν u x x {u_t} = u {u_{xx}} is solved by random walking the particles while u t = f ( u ) {u_t} = f(u) is solved with a numerical ordinary differential equation solver such as Euler’s method. We prove that the method converges when f ( u ) = u ( 1 − u ) f(u) = u(1 - u) , i.e. the Kolmogorov equation, and that when the time step Δ t \Delta t is O ( N 4 − 1 ) O({\sqrt [4]{N}^{ - 1}}) the rate of convergence is like ln ⁡ N ⋅ N 4 − 1 \ln N \cdot \,{\sqrt [4]{N}^{ - 1}} where N denotes the number of particles. Furthermore, we show that this rate of convergence is uniform as the diffusion coefficient ν u tends to 0. Thus, travelling waves with arbitrarily steep wavefronts may be modeled without an increase in the computational cost. We also present the results of numerical experiments including the use of second-order time discretization and second-order operator splitting and use these results to estimate the expected value and standard deviation of the error.