A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis

TL;DR

This paper introduces a new stochastic particle-field numerical method with spectral acceleration for 3D chemotaxis systems, achieving improved efficiency and accuracy, validated through rigorous error analysis and complex blowup dynamics simulations.

Contribution

The paper develops the SIPF-PIC method combining particle dynamics with spectral solvers, offering significant computational efficiency improvements and rigorous error estimates for 3D Keller-Segel systems.

Findings

Achieves $O(P + H^3 \log H)$ complexity per step

Validates convergence order through numerical experiments

Demonstrates ability to simulate complex blowup phenomena

Abstract

In this paper, we develop a novel numerical framework, namely the stochastic interacting particle-field method with particle-in-cell acceleration (SIPF-PIC), for the efficient simulation of the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) systems. The SIPF-PIC method integrates Lagrangian particle dynamics with spectral field solvers by leveraging localized particle-grid interpolations and fast Fourier transform (FFT) techniques. For $P$ particles and $H$ Fourier modes per spatial dimension, the SIPF-PIC method achieves a computational complexity of $O(P + H^3 \log H)$ per time step, a significant improvement over the original SIPF method (proposed in \cite{SIPF1}), which has a computational complexity of $O(PH^3)$, while preserving numerical accuracy. Moreover, we carry out a rigorous error analysis for the proposed method and establish the corresponding error estimates. Finally, we present numerical experiments to validate the convergence order and demonstrate the computational efficiency of SIPF-PIC. Further numerical experiments show the method's capability of capturing complex blowup dynamics beyond single-point collapse, including ring-shaped singularities.