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

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.