Error attribution in gradient random walk methods for parabolic equations

Abstract

Gradient random walk (GRW) methods represent a solution's spatial derivative by weighted particles and recover the solution by cumulative summation. We examine their accuracy for the one-dimensional heat, scalar cubic reaction-diffusion, and Burgers' equations. Thirty-seed ensembles, paired reconstructions, and deterministic controls identify the sources of two apparent accuracy limits. For the heat equation, comparing cumulative bin sums at bin centers introduces a first-order alignment error, which the exact reflected-particle distribution predicts and bin-edge comparison removes. An identity for the expected squared error predicts the remaining floor and the particle count for a target root-mean-square accuracy. With unequal weights, a count chosen in advance gives an error within 1% of its prediction. Reaction-front position, speed, and shape errors decrease under refinement. For Burgers' equation, exact transformed boundary data reduce the deterministic Cole-Hopf error about 490-fold, while fixed-endpoint particle runs approach their deterministic reference.

arXiv comment38 pages, 11 figures, 9 tables. Software and data are available at https://doi.org/10.5281/zenodo.22680376