Asymptotic preserving $P_N$ methods for haptotaxis equations

TL;DR

This paper develops an asymptotic preserving $P_N$ method for the haptotaxis equation, accurately capturing biological cell movement along tissue fibers with high-order precision in multiple dimensions.

Contribution

It introduces a generalized micro-macro decomposition for finite-volume schemes on staggered grids, enabling second-order accuracy and elimination of spurious modes in haptotaxis modeling.

Findings

Achieves up to second-order accuracy in space.

Effectively eliminates spurious modes in discretization.

Applicable to three-dimensional models of cell movement.

Abstract

The so-called haptotaxis equation is a special class of transport equation that arises from models of biological cell movement along tissue fibers. This equation has an anisotropic advection-diffusion equation as its macroscopic limit. An up to second-order accurate asymptotic preserving method is developed for the haptotaxis equation in space dimension up to three. For this the micro-macro decomposition proposed by Lemou and Mieussens is generalized in the context of finite-volume schemes on staggered grids. The spurious modes that arise from this discretization can be eliminated by combining flux evaluations from different points in the right way. The velocity space is discretized by an arbitrary-order linear moment system ($P_N$).