Numerical Approaches for non-local Transport-Dominated PDE Models with Applications to Biology

TL;DR

This paper reviews numerical methods for non-local transport-dominated PDE models in biology, highlighting challenges, recent schemes including a new semi-implicit method, and opportunities for advancing biological modeling.

Contribution

It provides a comprehensive overview of numerical approaches for non-local PDEs in biology and introduces a novel semi-implicit scheme to improve stability.

Findings

Finite element method offers promising solutions.

New semi-implicit scheme stabilizes oscillations.

Open problems remain in numerical simulation accuracy.

Abstract

Transport-dominated partial differential equation models have been used extensively over the past two decades to describe various collective migration phenomena in cell biology and ecology. To understand the behaviour of these models (and the biological systems they describe) different analytical and numerical approaches have been used. While the analytical approaches have been discussed by different recent review studies, the numerical approaches are still facing different open problems, and thus are being employed on a rather ad-hoc basis for each developed non-local model. The goal of this review is to summarise the basic ideas behind these transport-dominated non-local models, to discuss the current numerical approaches used to simulate these models, and finally to discuss some open problems related to the applications of these numerical methods, in particular the finite element method. This allows us to emphasize the opportunities offered by this numerical method to advance the research in this field. In addition, we present in detail some numerical schemes that we used to discretize these non-local equations; in particular a new semi-implicit scheme we introduced to stabilize the oscillations obtained with classical schemes.