Understanding differential equations through diffusion point of view: non-symmetric discrete equations

TL;DR

This paper introduces a diffusion-based approach using a modified D-iteration algorithm to efficiently solve non-symmetric discrete differential equations by decomposing the process per direction, applicable to problems solvable by Gauss-Seidel iteration.

Contribution

It adapts the D-iteration algorithm to handle non-symmetric discrete equations through a diffusion perspective, improving computational efficiency for certain differential problems.

Findings

Pre-computing diffusion for elementary catalysts enhances efficiency.

The method effectively decomposes diffusion processes per direction.

Applicable to problems solvable by Gauss-Seidel iteration.

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. It has been shown that pre-computing the diffusion process for an elementary catalyst case as a fundamental block of a class of differential equations, the computation efficiency can be greatly improved. Here, we explain how the diffusion point of view can be applied to decompose the fluid diffusion process per direction and how to handle non-symmetric discrete equations. 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.