Numerical Solution of linear drift-diffusion and pure drift equations on one-dimensional graphs

TL;DR

This paper develops finite volume numerical schemes with upwind flux and implicit time discretization for solving linear drift-diffusion and pure drift equations on one-dimensional graphs, ensuring positivity and uniqueness.

Contribution

It introduces a novel extension of finite volume schemes to complex graph geometries with bifurcations, providing theoretical guarantees and practical testing.

Findings

Discrete problems admit positive unique solutions

Methods successfully tested on electrical treeing geometry

Schemes handle arbitrary bifurcation nodes effectively

Abstract

We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite Volume schemes with upwind flux to domains presenting bifurcation nodes with an arbitrary number of incoming and outgoing edges, and implicit time discretization. We show that the discrete problems admit positive unique solutions, and we test the methods on the intricate geometry of an electrical treeing.