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High-order mass conserving, positivity plus energy-law preserving schemes and their error estimates for Keller-Segel equations

Mingmei Chen, Kun Wang, Cong Xie

TL;DR

This work tackles the numerical approximation of the Keller-Segel chemotaxis equations, which possess mass conservation, positivity, and an energy-dissipation property. It introduces a positivity-preserving reformulation via $u=\log(\rho)$ and develops high-order, linear, decoupled time discretizations using the backward differentiation formula $D_{k\tau}$, augmented with a recovery step for mass conservation and an energy-law preservation correction (EPC). The authors prove that the schemes achieve the optimal convergence rate $O(\tau^k)$ for $k=1\ldots 5$ under suitable regularity assumptions and mass constraints, while preserving mass, positivity, and the original energy law in practice. Numerical experiments corroborate the theory and highlight the schemes’ stability and physical fidelity, including robustness in blow-up regimes.

Abstract

Chemotaxis plays a significant role in numerous physiological processes. The Keller-Segel equation serves as a mathematical model for simulating the phenomenon of cell population aggregation under chemotaxis, possessing physical properties such as mass conservation, positivity of density, and energy dissipation. High-order linear and decoupled schemes for the parabolic-parabolic Keller-Segel chemotaxis model are proposed in this paper, which satisfy the three physical properties mentioned earlier. Firstly, by applying a logarithmic transformation, the Keller-Segel model is reformulated into its equivalent form that maintains the positivity of cell density regardless of the discrete scheme. Based on this equivalent system, we then propose high-order linear and decoupled numerical schemes using the backward differentiation formula (BDF). Furthermore, through the incorporation of a recovery technique and an energy-law preservation correction (EPC), we ensure that these schemes maintain mass conservation and preserve the original energy-law. Finally, we conduct a rigorous optimal error analysis for the numerical schemes under certain assumptions regarding the regularity of solutions, and some numerical experiments are also presented to demonstrate their effectiveness.

High-order mass conserving, positivity plus energy-law preserving schemes and their error estimates for Keller-Segel equations

TL;DR

This work tackles the numerical approximation of the Keller-Segel chemotaxis equations, which possess mass conservation, positivity, and an energy-dissipation property. It introduces a positivity-preserving reformulation via and develops high-order, linear, decoupled time discretizations using the backward differentiation formula , augmented with a recovery step for mass conservation and an energy-law preservation correction (EPC). The authors prove that the schemes achieve the optimal convergence rate for under suitable regularity assumptions and mass constraints, while preserving mass, positivity, and the original energy law in practice. Numerical experiments corroborate the theory and highlight the schemes’ stability and physical fidelity, including robustness in blow-up regimes.

Abstract

Chemotaxis plays a significant role in numerous physiological processes. The Keller-Segel equation serves as a mathematical model for simulating the phenomenon of cell population aggregation under chemotaxis, possessing physical properties such as mass conservation, positivity of density, and energy dissipation. High-order linear and decoupled schemes for the parabolic-parabolic Keller-Segel chemotaxis model are proposed in this paper, which satisfy the three physical properties mentioned earlier. Firstly, by applying a logarithmic transformation, the Keller-Segel model is reformulated into its equivalent form that maintains the positivity of cell density regardless of the discrete scheme. Based on this equivalent system, we then propose high-order linear and decoupled numerical schemes using the backward differentiation formula (BDF). Furthermore, through the incorporation of a recovery technique and an energy-law preservation correction (EPC), we ensure that these schemes maintain mass conservation and preserve the original energy-law. Finally, we conduct a rigorous optimal error analysis for the numerical schemes under certain assumptions regarding the regularity of solutions, and some numerical experiments are also presented to demonstrate their effectiveness.

Paper Structure

This paper contains 6 sections, 8 theorems, 103 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Schemes and main results
  4. Error Analysis
  5. Numerical Experiments
  6. Conclusions

Key Result

Theorem 3.1

Under the Assumptions as.2.1 and as.3.1, there exists a sufficiently small positive constant $\tau^*$ such that, for any $\tau$ satisfying $\tau\le\tau^*$ and for $n=k-1,\cdots,N-1$, the solutions obtained by the schemes eq.3.1-eq.3.6 fulfill

Figures (7)

  • Figure 1: Convergence rates in $L^2$-norm of the $k$th-order schemes.
  • Figure 2: Convergence rates in $H^1$-seminorm of the $k$th-order schemes.
  • Figure 3: Simulations of the density, mass and energy (Example 2).
  • Figure 4: Numerical approximations of $\rho(t_{N})$ and $c(t_{N})$ (Example 2).
  • Figure 5: The cross section of $\rho(x, 0, T)$ with $T=1\times10^{-4}$ (Example 3).
  • ...and 2 more figures

Theorems & Definitions (17)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Lemma 4.1: bib36, Lemma 1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 7 more