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Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations

Yubin Lu, Chi-An Chen, Xiaofan Li, Chun Liu

TL;DR

This paper develops two ADI-based finite difference schemes for the 2D Keller-Segel equations that preserve essential physical properties while reducing computational cost. The first-order-in-time ADI scheme with a two-step linear solve achieves positivity, mass conservation, and asymptotic energy dissipation, and is second-order in space; the second, a Crank-Nicolson–type additive ADI scheme, attains second-order accuracy in time but requires conditions for positivity. Numerical experiments confirm second-order spatial accuracy and (for the additive scheme) second-order temporal accuracy, while efficiency comparisons show substantial speedups over standard five-point discretizations. The methods provide scalable, structure-preserving tools for simulating chemotaxis models, including near-blow-up regimes, with practical implications for large-scale computations and longer-time integration.

Abstract

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.

Finite Difference Approximation with ADI Scheme for Two-dimensional Keller-Segel Equations

TL;DR

This paper develops two ADI-based finite difference schemes for the 2D Keller-Segel equations that preserve essential physical properties while reducing computational cost. The first-order-in-time ADI scheme with a two-step linear solve achieves positivity, mass conservation, and asymptotic energy dissipation, and is second-order in space; the second, a Crank-Nicolson–type additive ADI scheme, attains second-order accuracy in time but requires conditions for positivity. Numerical experiments confirm second-order spatial accuracy and (for the additive scheme) second-order temporal accuracy, while efficiency comparisons show substantial speedups over standard five-point discretizations. The methods provide scalable, structure-preserving tools for simulating chemotaxis models, including near-blow-up regimes, with practical implications for large-scale computations and longer-time integration.

Abstract

Keller-Segel systems are a set of nonlinear partial differential equations used to model chemotaxis in biology. In this paper, we propose two alternating direction implicit (ADI) schemes to solve the 2D Keller-Segel systems directly with minimal computational cost, while preserving positivity, energy dissipation law and mass conservation. One scheme unconditionally preserves positivity, while the other does so conditionally. Both schemes achieve second-order accuracy in space, with the former being first-order accuracy in time and the latter second-order accuracy in time. Besides, the former scheme preserves the energy dissipation law asymptotically. We validate these results through numerical experiments, and also compare the efficiency of our schemes with the standard five-point scheme, demonstrating that our approaches effectively reduce computational costs.
Paper Structure (18 sections, 5 theorems, 100 equations, 5 figures, 5 tables)

This paper contains 18 sections, 5 theorems, 100 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. ADI schemes
  3. The ADI scheme
  4. Conservative properties of the ADI scheme
  5. Positivity preservation
  6. Conservation of mass
  7. Energy dissipation
  8. A second-order scheme
  9. Numerical experiments
  10. The order of convergence
  11. Second-order convergence in time
  12. Efficiency
  13. An illustrative example
  14. Conclusion
  15. Detailed Expressions for the Operators $\tau_x,\tau_y$ and $\tau_x\tau_y$
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $\rho_{i,j}^{0}\geq 0$, then the ADI scheme eqn:semi-discrete4ADI_rho2 preserves $\rho_{i,j}^{n}\geq 0, n\in\mathbb{N}$.

Figures (5)

  • Figure 1: The evolution of the minimum values of the density $\rho$ and the concentration $c$ for the solution of the Keller-Segel equations \ref{['eqn:original_rho']} and \ref{['eqn:original_c']} with $\epsilon=1$, the initial condition \ref{['eqn:icie1']} and the periodic or vanishing Neumann boundary conditions. (Left) $\rho_{\text{min}}:=\min\limits_{i,j}\{\rho_{i,j}\}$. (Right) $c_{\text{min}}:=\min\limits_{i,j}\{c_{i,j}\}$.
  • Figure 2: The quantities defined in \ref{['eqn:mass-tau_x']} and \ref{['eqn:mass-tau_xy']} for the solution of the Keller-Segel equations \ref{['eqn:original_rho']} and \ref{['eqn:original_c']} with $\epsilon=1$ and the initial condition \ref{['eqn:icie1']}. (Left) Periodic boundary conditions. (Right) Vanishing Neumann boundary conditions.
  • Figure 3: The evolution of the total mass compared with the initial values, namely $\rho_{\text{tot}} (t) - \rho_{\text{tot}}(0)$ and $c_{\text{tot}} (t) - c_{\text{tot}}(0) - \rho_{\text{tot}}(0) t$, for the solution of the Keller-Segel equations \ref{['eqn:original_rho']} and \ref{['eqn:original_c']} with $\epsilon=1$ and the initial condition \ref{['eqn:icie1']}. (Left) Periodic boundary conditions. (Right) Vanishing Neumann boundary conditions.
  • Figure 4: (Left) The evolution of the discrete free energy $\mathcal{E}^n$\ref{['eqn:fully_discrete_energy_1']} for the solution of the Keller-Segel equations \ref{['eqn:original_rho']} and \ref{['eqn:original_c']} with $\epsilon=1$ and the initial condition \ref{['eqn:icie1']} with periodic boundary conditions. (Right) The corresponding the energy decay rate $(\mathcal{E}^{n+1}-\mathcal{E}^n)/\Delta t$ and the RHS of \ref{['eqn:fullyenergy']} divided by $\Delta t$.
  • Figure 5: (Left) The evolution of the discrete free energy $\mathcal{E}^n$\ref{['eqn:fully_discrete_energy_1']} for the solution of the Keller-Segel equations \ref{['eqn:original_rho']} and \ref{['eqn:original_c']} with $\epsilon=1$ and the initial condition \ref{['eqn:icie1']} with vanishing Neumann boundary conditions. (Right) The corresponding the energy decay rate $(\mathcal{E}^{n+1}-\mathcal{E}^n)/\Delta t$ and the RHS of \ref{['eqn:fullyenergy']} divided by $\Delta t$.

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.2
  • proof
  • Proposition 2.1
  • proof
  • Remark 2.3
  • Proposition 2.2
  • ...and 9 more