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Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial Systems

Jie Xu, Shusen Xie, Hongfei Fu

TL;DR

The paper tackles a nonlinear diffusion–reaction system modeling bacterial spread and disease transmission by developing high-order, linearized CN-based schemes. It introduces two auxiliary variables to reformulate the problem and constructs a linearized, decoupled fourth-order compact finite difference method (CN-CFD) with unconditional maximum-norm stability and $O(\tau^2 + h_x^4 + h_y^4)$ convergence. To boost efficiency, the authors further devise a compact CN-ADI-CFD scheme that reduces the 2D problem to 1D subproblems while preserving stability and optimal max-norm accuracy. Rigorous energy-based and temporal-spatial error-splitting analyses underpin the stability and convergence results, and extensive numerical experiments validate the theoretical rates and demonstrate extinction and endemic equilibrium phenomena. The work provides highly accurate, stable, and computationally efficient tools for simulating nonlinear bacterial systems and related epidemic dynamics on two-dimensional domains.

Abstract

In this paper, by introducing two temporal-derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The temporal-spatial error splitting technique and discrete energy method are employed to prove the unconditional stability and convergence of the method in discrete maximum norm. Furthermore, to improve the computational efficiency, an alternating direction implicit (ADI) compact difference algorithm is proposed, and the unconditional stability and optimal-order maximum-norm error estimate for the ADI scheme are also strictly established. Finally, several numerical experiments are conducted to validate the theoretical convergence and to simulate the phenomena of bacterial extinction as well as the formation of endemic diseases.

Maximum-Norm Error Estimates of Fourth-Order Compact and ADI Compact Finite Difference Methods for Nonlinear Coupled Bacterial Systems

TL;DR

The paper tackles a nonlinear diffusion–reaction system modeling bacterial spread and disease transmission by developing high-order, linearized CN-based schemes. It introduces two auxiliary variables to reformulate the problem and constructs a linearized, decoupled fourth-order compact finite difference method (CN-CFD) with unconditional maximum-norm stability and convergence. To boost efficiency, the authors further devise a compact CN-ADI-CFD scheme that reduces the 2D problem to 1D subproblems while preserving stability and optimal max-norm accuracy. Rigorous energy-based and temporal-spatial error-splitting analyses underpin the stability and convergence results, and extensive numerical experiments validate the theoretical rates and demonstrate extinction and endemic equilibrium phenomena. The work provides highly accurate, stable, and computationally efficient tools for simulating nonlinear bacterial systems and related epidemic dynamics on two-dimensional domains.

Abstract

In this paper, by introducing two temporal-derivative-dependent auxiliary variables, a linearized and decoupled fourth-order compact finite difference method is developed and analyzed for the nonlinear coupled bacterial systems. The temporal-spatial error splitting technique and discrete energy method are employed to prove the unconditional stability and convergence of the method in discrete maximum norm. Furthermore, to improve the computational efficiency, an alternating direction implicit (ADI) compact difference algorithm is proposed, and the unconditional stability and optimal-order maximum-norm error estimate for the ADI scheme are also strictly established. Finally, several numerical experiments are conducted to validate the theoretical convergence and to simulate the phenomena of bacterial extinction as well as the formation of endemic diseases.
Paper Structure (21 sections, 13 theorems, 135 equations, 8 figures, 3 tables)

This paper contains 21 sections, 13 theorems, 135 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Some notations and auxiliary lemmas
  3. A fourth-order compact difference scheme and its unconditional analysis
  4. Construction of compact difference scheme
  5. Analysis of the CN-CFD scheme
  6. Step I. Estimates for $\|\mathcal{H}_h s^{n}\|$ and $\|\mathcal{H}_h e^{n}\|$
  7. Step II. Estimates for $\|\mathcal{H}_h l^{n}\|$ and $\|\mathcal{H}_h w^{n}\|$
  8. Step III. Estimates for $\|s^{m}\|_\infty$ and $\|l^{m}\|_\infty$
  9. Result of Step I
  10. Result of Step II
  11. Result of Step III
  12. A compact ADI algorithm and its numerical analysis
  13. Derivation of the compact ADI scheme
  14. Analysis of the compact ADI scheme
  15. Step I. Estimates for $\| \mathcal{H}_h s^n \|$ and $\left\| \Lambda_h s^n\right\|$
  16. ...and 6 more sections

Key Result

Lemma 2.1

Let $f(t) \in C^1([t_{0}, t_{1/2}])$, then we have

Figures (8)

  • Figure 1: Extinction phenomenon of the average concentration of bacteria $u$.
  • Figure 2: Extinction phenomenon of the average concentration of infective people $v$.
  • Figure 3: Endemic phenomenon of the average concentration of bacteria $u$.
  • Figure 4: Endemic phenomenon of the average concentration of infective people $v$.
  • Figure 5: Evolution of $u$ with parameters (a) in Example \ref{['exam:compact']} and initial conditions $(u_0^-,v_0^-)$ (up) and $(u_0^+,v_0^+)$ (down).
  • ...and 3 more figures

Theorems & Definitions (23)

  • Lemma 2.1: SZG
  • Lemma 2.2: SZG
  • Lemma 2.3: SZG
  • Lemma 2.4: BH
  • Lemma 2.5: XYK
  • Lemma 2.6: LSS
  • Lemma 2.7
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 13 more