Convergence of the EBT method for a non-local model of cell proliferation with discontinuous interaction kernel

TL;DR

This paper proves the convergence of the EBT particle method for a non-local cell proliferation model with a discontinuous interaction kernel by transforming the problem into a radial coordinate system and introducing a new weighted norm.

Contribution

It introduces a novel approach using spherical coordinates and a weighted flat norm to establish convergence for a non-Lipschitz kernel in a non-local PDE.

Findings

Convergence of the particle method in the new weighted flat norm.

Numerical simulations support the theoretical convergence results.

Extension discussion for the two-dimensional case.

Abstract

We consider the EBT algorithm (a particle method) for the non-local equation with a discontinuous interaction kernel. The main difficulty lies in the low regularity of the kernel which is not Lipschitz continuous, thus preventing the application of standard arguments. Therefore, we use the radial symmetry of the problem instead and transform it using spherical coordinates. The resulting equation has a Lipschitz kernel with only one singularity at zero. We introduce a new weighted flat norm and prove that the particle method converges in this norm. We also comment on the two-dimensional case which requires the application of the theory of measure spaces on general metric spaces and present numerical simulations confirming the theoretical results. In a companion paper, we apply the Bayesian method to fit parameters to this model and study its theoretical properties.