On the Accuracy of Gradient Random Walk Methods for the Heat, FitzHugh-Nagumo, and Burgers' Equations

Abstract

Gradient Random Walk (GRW) methods represent the spatial derivative of a solution with weighted particles and recover the solution by cumulative summation. Measured accuracy depends not only on the particle count but also on where the reconstruction is evaluated, how the boundary data are incorporated, and how the physical solution is recovered from the computed field. We separate these contributions for the heat equation, a scalar FitzHugh-Nagumo traveling front, and Burgers' equation treated through the Cole-Hopf transformation, using multi-seed ensembles, paired reconstructions of identical trajectories, and deterministic controls that distinguish stochastic from systematic error. For the heat equation, an apparent error plateau at fixed bin count is traced to a half-bin mismatch between the cumulative sum and its comparison points, and realigning the comparison removes it. For Burgers' equation, the accuracy of the recovered solution is set by the boundary data for the transformed variable, and exact transformed data remove this limit. For the FitzHugh-Nagumo front, errors in the profile, front location, and speed decrease under particle refinement, before and after the translational component is removed, verifying the deterministic reaction-weight formulation. With the evaluation and boundary conventions held fixed, the stochastic error decreases in the particle count $N$, consistent with the Monte Carlo convergence rate $O(N^{-1/2})$. These findings identify the operations that govern measured GRW accuracy and show how to improve it.