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A second-order accurate, original energy dissipative numerical scheme for chemotaxis and its convergence analysis

Jie Ding, Cheng Wang, Shenggao Zhou

TL;DR

This work develops a second-order in time, variationally based numerical scheme for the Patlak–Keller–Segel chemotaxis model with non-constant mobilities, achieving positivity preservation and original energy dissipation at the discrete level. A modified Crank–Nicolson discretization of the entropy term, together with an extrapolated mobility and a Taylor-expanded logarithmic term, yields a convex discrete energy minimization problem with unique solvability. The authors establish a rigorous convergence analysis, including higher-order consistency, rough and refined error estimates, and recovery of a-priori bounds, proving an optimal rate of convergence under a linear refinement constraint. Numerical experiments in 2D confirm second-order accuracy and demonstrate robust preservation of mass, energy dissipation, and positivity during blowup scenarios for symmetric and asymmetric initial data. The results provide a reliable, structure-preserving framework for simulating chemotaxis with complex mobilities and offer a path toward accurate, physically consistent simulations of aggregation phenomena.

Abstract

This paper proposes a second-order accurate numerical scheme for the Patlak-Keller-Segel system with various mobilities for the description of chemotaxis. Formulated in a variational structure, the entropy part is novelly discretized by a modified Crank-Nicolson approach so that the solution to the proposed nonlinear scheme corresponds to a minimizer of a convex functional. A careful theoretical analysis reveals that the unique solvability and positivity-preserving property could be theoretically justified. More importantly, such a second order numerical scheme is able to preserve the dissipative property of the original energy functional, instead of a modified one. To the best of our knowledge, the proposed scheme is the first second-order accurate one in literature that could achieve both the numerical positivity and original energy dissipation. In addition, an optimal rate convergence estimate is provided for the proposed scheme, in which rough and refined error estimate techniques have to be included to accomplish such an analysis. Ample numerical results are presented to demonstrate robust performance of the proposed scheme in preserving positivity and original energy dissipation in blowup simulations.

A second-order accurate, original energy dissipative numerical scheme for chemotaxis and its convergence analysis

TL;DR

This work develops a second-order in time, variationally based numerical scheme for the Patlak–Keller–Segel chemotaxis model with non-constant mobilities, achieving positivity preservation and original energy dissipation at the discrete level. A modified Crank–Nicolson discretization of the entropy term, together with an extrapolated mobility and a Taylor-expanded logarithmic term, yields a convex discrete energy minimization problem with unique solvability. The authors establish a rigorous convergence analysis, including higher-order consistency, rough and refined error estimates, and recovery of a-priori bounds, proving an optimal rate of convergence under a linear refinement constraint. Numerical experiments in 2D confirm second-order accuracy and demonstrate robust preservation of mass, energy dissipation, and positivity during blowup scenarios for symmetric and asymmetric initial data. The results provide a reliable, structure-preserving framework for simulating chemotaxis with complex mobilities and offer a path toward accurate, physically consistent simulations of aggregation phenomena.

Abstract

This paper proposes a second-order accurate numerical scheme for the Patlak-Keller-Segel system with various mobilities for the description of chemotaxis. Formulated in a variational structure, the entropy part is novelly discretized by a modified Crank-Nicolson approach so that the solution to the proposed nonlinear scheme corresponds to a minimizer of a convex functional. A careful theoretical analysis reveals that the unique solvability and positivity-preserving property could be theoretically justified. More importantly, such a second order numerical scheme is able to preserve the dissipative property of the original energy functional, instead of a modified one. To the best of our knowledge, the proposed scheme is the first second-order accurate one in literature that could achieve both the numerical positivity and original energy dissipation. In addition, an optimal rate convergence estimate is provided for the proposed scheme, in which rough and refined error estimate techniques have to be included to accomplish such an analysis. Ample numerical results are presented to demonstrate robust performance of the proposed scheme in preserving positivity and original energy dissipation in blowup simulations.
Paper Structure (19 sections, 12 theorems, 135 equations, 5 figures)

This paper contains 19 sections, 12 theorems, 135 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Chemotaxis models
  3. The numerical scheme
  4. Some notations
  5. Second-order accurate numerical scheme
  6. Structure-preserving properties
  7. Convergence analysis
  8. Higher-order consistency analysis
  9. A rough error estimate
  10. A refined error estimate
  11. Recovery of the a-priori assumption (\ref{['A-priAssum:eq1']})
  12. Numerical results
  13. Accuracy test
  14. Performance test
  15. Symmetric initial data on a square
  16. ...and 4 more sections

Key Result

Lemma 3.1

wang11awise09aguo16 For any $\phi_1$, $\phi_2$, $\phi_3$, $g\in\mathcal{C}$, and any $\vec{f} = (f^x,f^y,f^z)$, with $f^x$, $f^y$ and $f^z$ evaluated at $(i+1/2,j,k), (i,j+1/2,k), (i,j,k+1/2)$, respectively, the following summation-by-parts formulas are valid: in which $\mathcal{A}_h$ corresponds to the average operator given by $A_x$, $A_y$, and $A_z$.

Figures (5)

  • Figure 1: Numerical errors (in $\ell^\infty$) of $\rho$ and $\phi$ computed by the second-order accurate scheme (\ref{['PKS2nd_1']}-\ref{['PKS2nd_2']}) at a final time $T=0.1$, with a mesh ratio $\Delta t=h/10$. Various values of $\theta$ are considered: $\theta=1$, $\theta=1e-2$, and $\theta=1e-4$.
  • Figure 2: Evolution of density $\rho$ at time instants: $T=0$, $0.12$, $0.15$, and $0.16$, with $\Delta t=10^{-5}$ and $h=10^{-2}$.
  • Figure 3: Time evolution of the discrete energy $F_h$, mass of $\rho$, and the maximum and minimum values of $\rho$ over the computational mesh, with $\Delta t=10^{-5}$.
  • Figure 4: Time evolution of $\rho$ at a sequence of time instants: $T=0$, $0.04$, $0.076$, and $0.08$, with $\Delta t=10^{-5}$ and $h=1/100$.
  • Figure 5: Time evolution of the discrete energy $F_h$, mass of $\rho$, and the minimum and maximum values of $\rho$ over the computational mesh, with $\Delta t=10^{-5}$.

Theorems & Definitions (15)

  • Lemma 3.1
  • Remark 3.2
  • Theorem 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Remark 4.6
  • Remark 4.7
  • Theorem 5.1
  • ...and 5 more