Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance

TL;DR

This paper analyzes particle methods for structured population models with nonlocal boundary terms, establishing well-posedness and convergence using a metric space framework, and compares two algorithms through numerical simulations.

Contribution

It introduces a rigorous analysis of particle methods for structured population models, proving well-posedness and convergence rates within a metric space framework.

Findings

Both algorithms are shown to be well-posed and convergent.

Numerical simulations validate theoretical convergence rates.

Distances between solutions are effectively measured using the flat metric.

Abstract

Recently developed theoretical framework for analysis of structured population dynamics in the spaces of nonnegative Radon measures with a suitable metric provides a rigorous tool to study numerical schemes based on particle methods. The approach is based on the idea of tracing growth and transport of measures which approximate the solution of original partial differential equation. In this paper we present analytical and numerical study of two versions of Escalator Boxcar Train (EBT) algorithm which has been widely applied in theoretical biology, and compare it to the recently developed split-up algorithm. The novelty of this paper is in showing well-posedness and convergence rates of the schemes using the concept of semiflows on metric spaces. Theoretical results are validated by numerical simulations of test cases, in which distances between simulated and exact solutions are computed using flat metric.