Complete flux scheme for variable velocity fields: coupling between the advection-diffusion equation and the Poisson equation for the velocity field

TL;DR

This paper extends the complete flux scheme to coupled advection-diffusion and Poisson equations, ensuring second-order convergence across Péclet numbers by approximating the velocity field with a piecewise linear approach.

Contribution

It introduces a novel coupling method that preserves second-order accuracy in complex models involving variable velocity fields and source terms.

Findings

Achieved uniform second-order convergence in Péclet numbers.

Extended the complete flux scheme to coupled advection-diffusion and Poisson equations.

Demonstrated effectiveness in plasma physics and porous media models.

Abstract

In this work, we consider an advection-diffusion equation, coupled to a Poisson equation for the velocity field. This type of coupling is typically encountered in models arising from plasma physics or porous media flow. The aim of this work is to build upon the complete flux scheme (an improvement over the Scharfetter-Gummel scheme by considering the contribution of the source term), so that its second-order convergence, which is uniform in P\'eclet numbers, carries over to these models. This is done by considering a piecewise linear approximation of the velocity field, which is then used for defining upwind-adjusted P\'eclet numbers.