Understanding differential equations through diffusion point of view

TL;DR

This paper introduces a novel diffusion-based adaptation of the D-iteration algorithm to efficiently solve differential equations by modeling them as diffusion processes, enhancing computational performance for certain classes of problems.

Contribution

It presents a new diffusion point of view and an adapted D-iteration method for solving differential equations more efficiently, especially those suitable for Gauss-Seidel iteration.

Findings

Pre-computing diffusion for elementary catalysts improves efficiency.

The method applies to problems solvable by Gauss-Seidel iteration.

Significant computational speed-up demonstrated.

Abstract

In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the boundary or initial conditions are replaced by fluid catalysts. Pre-computing the diffusion process for an elementary catalyst case as a fundamental block of a class of differential equations, we show that the computation efficiency can be greatly improved. The method can be applied on the class of problems that can be addressed by the Gauss-Seidel iteration, based on the linear approximation of the differential equations.