Positivity preserving high order schemes for angiogenesis models

TL;DR

This paper develops high-order positivity preserving numerical schemes for angiogenesis models, accurately capturing blood vessel formation and spread in hypoxic tissues through advanced discretization techniques.

Contribution

It introduces a novel combination of asymptotic reduction with WENO and SSP time discretization for improved angiogenesis simulations.

Findings

Successfully captures soliton-like blood vessel formation

Maintains positivity and high order accuracy in simulations

Demonstrates effectiveness in modeling angiogenesis dynamics

Abstract

Hypoxy induced angiogenesis processes can be described coupling an integrodifferential kinetic equation of Fokker-Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We show that soliton-like solutions representing blood vessel formation and spread towards hypoxic regions are captured.