Initial-Boundary Value Problem for the heat equation - A stochastic algorithm

TL;DR

This paper introduces a novel stochastic algorithm based on random walks on heat balls to solve initial-boundary value problems for the heat equation, demonstrating efficiency and accuracy through theoretical and numerical analysis.

Contribution

It presents a new algorithm using heat ball-based random walks and a mean value formula, extending the Walk on Spheres method to heat equations.

Findings

Algorithm is easy to implement.

Convergence is theoretically proven.

Numerical examples show high accuracy.

Abstract

The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to solve the Dirich-let problem for Laplace's equation, its implementation is rather easy. The definition of the random walk is based on a new mean value formula for the heat equation. The convergence results and numerical examples permit to emphasize the efficiency and accuracy of the algorithm.