A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis
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| Format: | Preprint |
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2025
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| _version_ | 1866918329932840960 |
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| author | Hu, Jingyuan Wang, Zhongjian Xin, Jack Zhang, Zhiwen |
| author_facet | Hu, Jingyuan Wang, Zhongjian Xin, Jack Zhang, Zhiwen |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_03452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis Hu, Jingyuan Wang, Zhongjian Xin, Jack Zhang, Zhiwen Numerical Analysis 35K55, 65M12, 65M70, 65M75, 65Y20 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. |
| title | A fast stochastic interacting particle-field method for 3D parabolic parabolic Chemotaxis systems: numerical algorithms and error analysis |
| topic | Numerical Analysis 35K55, 65M12, 65M70, 65M75, 65Y20 |
| url | https://arxiv.org/abs/2512.03452 |