Numerical scheme for kinetic transport equation with internal state *

TL;DR

This paper develops and analyzes numerical schemes for a two-stream kinetic system with an internal state modeling cell chemotaxis, ensuring accuracy across different scaling regimes and validating with simulations.

Contribution

It introduces uniformly accurate numerical schemes for a kinetic chemotaxis model with internal state, bridging microscopic and macroscopic descriptions.

Findings

Schemes converge to limiting macroscopic or kinetic systems.

Numerical simulations validate the accuracy and consistency.

Comparison with Monte Carlo confirms effectiveness.

Abstract

We investigate the numerical discretization of a two-stream kinetic system with an internal state, such system has been introduced to model the motion of cells by chemotaxis. This internal state models the intracellular methylation level. It adds a variable in the mathematical model, which makes it more challenging to simulate numerically. Moreover, it has been shown that the macroscopic or mesoscopic quantities computed from this system converge to the Keller-Segel system at diffusive scaling or to the velocity-jump kinetic system for chemotaxis at hyperbolic scaling. Then we pay attention to propose numerical schemes uniformly accurate with respect to the scaling parameter. We show that these schemes converge to some limiting schemes which are consistent with the limiting macroscopic or kinetic system. This study is illustrated with some numerical simulations and comparisons with Monte Carlo simulations.