Bound-Preserving Finite-Volume Schemes for Systems of Continuity Equations with Saturation

TL;DR

This paper introduces finite-volume schemes for continuity equations with saturation effects that preserve positivity, bounds, and energy dissipation, applicable to coupled systems in physics and biology, validated through extensive simulations.

Contribution

The paper develops a general finite-volume framework that maintains bounds and energy dissipation for nonlinear coupled continuity systems with saturation effects.

Findings

Schemes preserve positivity and bounds in simulations.

Energy dissipation is maintained at the discrete level.

New biological phenomena are demonstrated through simulations.

Abstract

We propose finite-volume schemes for general continuity equations which preserve positivity and global bounds that arise from saturation effects in the mobility function. In the case of gradient flows, the schemes dissipate the free energy at the fully discrete level. Moreover, these schemes are generalised to coupled systems of non-linear continuity equations, such as multispecies models in mathematical physics or biology, preserving the bounds and the dissipation of the energy whenever applicable. These results are illustrated through extensive numerical simulations which explore known behaviours in biology and showcase new phenomena not yet described by the literature.