A second order scheme for a Robin boundary condition in random walk algorithms

TL;DR

This paper introduces a second order numerical scheme for implementing Robin boundary conditions in Random Walk algorithms, significantly reducing computational errors without extra cost.

Contribution

The work develops a second order scheme for Robin boundary conditions in Random Walk models, improving accuracy over first order methods.

Findings

Second order scheme reduces computational error significantly.

The new scheme achieves higher accuracy without additional computational cost.

Validation against analytical and numerical solutions confirms improved performance.

Abstract

Random Walk (RW) is a common numerical tool for modeling the Advection-Diffusion equation. In this work, we develop a second order scheme for incorporating a heterogeneous reaction (i.e., a Robin boundary condition) in the RW model. In addition, we apply the approach in two test cases. We compare the second order scheme with the first order one as well as with analytical and other numerical solution. We show that the new scheme can reduce the computational error significantly, relative to the first order scheme. This reduction comes at no additional computational cost.