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A Hybrid Finite-Difference-Particle Method for Chemotaxis Models

Alina Chertock, Shumo Cui, Alexander Kurganov, Chenxi Wang

TL;DR

The paper addresses finite-time blowup phenomena in the two-species Patlak-Keller-Segel chemotaxis system and introduces a hybrid finite-difference–particle (FDP) method to accurately resolve singular structures, particularly when the two species blow up at different rates.It combines a sticky particle method for the density equations with a second-order finite-difference scheme for the chemoattractant and employs projection operators to couple the particle and grid representations.Key contributions include two merger steps for robust spike resolution, a variable diffusion treatment to maintain accuracy near concentration points, and a rigorous demonstration that the method captures both delta-type and algebraic blowup patterns in one- and two-species settings.Numerical experiments on parabolic-parabolic and parabolic-elliptic couplings show sharp blowup resolution with relatively few particles and improvements over grid-based methods in capturing the intricate dynamics of chemotactic aggregation.

Abstract

Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups. Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation. In this paper, we propose a hybrid finite-difference-particle (FDP) method, in which a particle method is used to solve the chemotaxis equation(s), while finite difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.

A Hybrid Finite-Difference-Particle Method for Chemotaxis Models

TL;DR

The paper addresses finite-time blowup phenomena in the two-species Patlak-Keller-Segel chemotaxis system and introduces a hybrid finite-difference–particle (FDP) method to accurately resolve singular structures, particularly when the two species blow up at different rates.It combines a sticky particle method for the density equations with a second-order finite-difference scheme for the chemoattractant and employs projection operators to couple the particle and grid representations.Key contributions include two merger steps for robust spike resolution, a variable diffusion treatment to maintain accuracy near concentration points, and a rigorous demonstration that the method captures both delta-type and algebraic blowup patterns in one- and two-species settings.Numerical experiments on parabolic-parabolic and parabolic-elliptic couplings show sharp blowup resolution with relatively few particles and improvements over grid-based methods in capturing the intricate dynamics of chemotactic aggregation.

Abstract

Chemotaxis systems play a crucial role in modeling the dynamics of bacterial and cellular behaviors, including propagation, aggregation, and pattern formation, all under the influence of chemical signals. One notable characteristic of these systems is their ability to simulate concentration phenomena, where cell density undergoes rapid growth near specific concentration points or along certain curves. Such growth can result in singular, spiky structures and lead to finite-time blowups. Our investigation focuses on the dynamics of the Patlak-Keller-Segel chemotaxis system and its two-species extensions. In the latter case, different species may exhibit distinct chemotactic sensitivities, giving rise to very different rates of cell density growth. Such a situation may be extremely challenging for numerical methods as they may fail to accurately capture the blowup of the slower-growing species mainly due to excessive numerical dissipation. In this paper, we propose a hybrid finite-difference-particle (FDP) method, in which a particle method is used to solve the chemotaxis equation(s), while finite difference schemes are employed to solve the chemoattractant equation. Thanks to the low-dissipation nature of the particle method, the proposed hybrid scheme is particularly adept at capturing the blowup behaviors in both one- and two-species cases. The proposed hybrid FDP methods are tested on a series of challenging examples, and the obtained numerical results demonstrate that our hybrid method can provide sharp resolution of the singular structures even with a relatively small number of particles. Moreover, in the two-species case, our method adeptly captures the blowing-up solution for the component with lower chemotactic sensitivity, a feature not observed in other works.
Paper Structure (15 sections, 33 equations, 6 figures, 1 table)

This paper contains 15 sections, 33 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Weighted Particle Method: an Overview
  3. Hybrid FDP Method for (\ref{['KS']})
  4. Sticky Particle Method for Equations (\ref{['KS1']}) and (\ref{['KS2']})
  5. Merger Step 1.
  6. Time Evolution.
  7. Merger Step 2.
  8. Finite-Difference Scheme for Equation (\ref{['KS3']})
  9. Projection Between the Particle and Grid Data
  10. Computation of $\nabla c$ and $\Delta c$ at Particle Locations
  11. Computation of $\rho_1$ and $\rho_2$ at Grid Points
  12. Numerical Examples
  13. Parabolic-Parabolic Case ($\tau=1$)
  14. Parabolic-Elliptic Case ($\tau=0$)
  15. Conclusion

Figures (6)

  • Figure 4.1: Example 2: $\rho_1(\bm x,0.0005)$ (upper left), $\rho_2(\bm x,0.0005)$ (upper right), $\rho_1(\bm x,0.001)$ (lower left), and $\rho_2(\bm x,0.001)$ (lower right), obtained using the hybrid FDP method with $\Delta=1/20$.
  • Figure 4.2: Example 3: $\rho(\bm x,0.02)$ (upper left), $\rho(\bm x,0.05)$ (upper right), $\rho(\bm x,0.1)$ (lower left), and $\rho(\bm x,0.2)$ (lower right), obtained using the hybrid FDP method with $\Delta=1/20$.
  • Figure 4.3: Example 3: particle locations at $t=0.02$ (left) and $t=0.1$ (right), obtained using the hybrid FDP method with $\Delta=1/20$. The color indicates the value of $w_i(t)/|\Omega_i(t)|$.
  • Figure 4.4: Example 4: $\rho(\bm x,0.02)$ (upper left), $\rho(\bm x,0.05)$ (upper right), $\rho(\bm x,0.1)$ (lower left), and the location of the "dominating" particle at $t=0.1$ (lower right), obtained using the hybrid FDP method with $\Delta=1/20$.
  • Figure 4.5: Example 5: $\rho_1(\bm x,0.003)$ (upper left), $\rho_2(\bm x,0.003)$ (upper right), $\rho_1(\bm x,0.0033)$ (lower left), and $\rho_2(\bm x,0.0033)$ (lower right), obtained using the hybrid FDP method with $\Delta=1/20$.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4