Long-time behaviour of hybrid finite volume schemes for advection-diffusion equations: linear and nonlinear approaches

TL;DR

This paper investigates the long-time behavior of hybrid finite volume schemes for linear and nonlinear advection-diffusion equations, proving exponential convergence to steady states and demonstrating positivity and accuracy through numerical simulations.

Contribution

It introduces a new nonlinear hybrid finite volume scheme and proves its convergence and positivity, extending the analysis to general polygonal meshes.

Findings

Discrete solutions converge exponentially fast to steady states

The nonlinear scheme maintains positivity of solutions

Numerical tests confirm theoretical long-time behavior and accuracy

Abstract

We are interested in the long-time behaviour of approximate solutions to heterogeneous and anisotropic linear advection-diffusion equations in the framework of hybrid finite volume (HFV) methods on general polygonal/polyhedral meshes. We consider two linear methods, as well as a new, nonlinear scheme, for which we prove the existence and the positivity of discrete solutions. We show that the discrete solutions to the three schemes converge exponentially fast in time towards the associated discrete steady-states. To illustrate our theoretical findings, we present some numerical simulations assessing long-time behaviour and positivity. We also compare the accuracy of the schemes on some numerical tests in the stationary case.