Understanding differential equations through diffusion point of view
This work offers an incremental improvement in numerical methods for solving differential equations, particularly for problems amenable to Gauss-Seidel iteration.
The authors propose a new adaptation of the D-iteration algorithm for numerically solving differential equations by reinterpreting the problem as a diffusion process with fluid catalysts, achieving improved computational efficiency.
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.