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A finite volume method preserving the invariant region property for the quasimonotone reaction-diffusion systems

Huifang Zhou

TL;DR

This work addresses preserving the invariant region for two-component quasimonotone reaction–diffusion systems on general polygonal meshes. It develops an IRP-preserving finite volume scheme using a DMP-preserving flux and backward-Euler time stepping, together with an IRP-preserving iterative solver for the nonlinear system. It proves IRP preservation under Lipschitz and quasimonotone assumptions, with explicit time-step constraints for the nonlinear solve, and demonstrates unconditional IRP for the iterative method. Numerical experiments on distorted meshes confirm second-order spatial accuracy and robust IRP preservation, outperforming a nine-point scheme that fails to maintain the invariant region.

Abstract

We present a finite volume method preserving the invariant region property (IRP) for the reaction-diffusion systems with quasimonotone functions, including nondecreasing, decreasing, and mixed quasimonotone systems. The diffusion terms and time derivatives are discretized by a finite volume method satisfying the discrete maximum principle (DMP) and the backward Euler method, respectively. The discretization leads to an implicit and nonlinear scheme, and it is proved to preserve the invariant region property unconditionally. We construct an iterative algorithm and prove the invariant region property ar each iteration step. Numerical examples are shown to confirm the accuracy and invariant region property of our scheme.

A finite volume method preserving the invariant region property for the quasimonotone reaction-diffusion systems

TL;DR

This work addresses preserving the invariant region for two-component quasimonotone reaction–diffusion systems on general polygonal meshes. It develops an IRP-preserving finite volume scheme using a DMP-preserving flux and backward-Euler time stepping, together with an IRP-preserving iterative solver for the nonlinear system. It proves IRP preservation under Lipschitz and quasimonotone assumptions, with explicit time-step constraints for the nonlinear solve, and demonstrates unconditional IRP for the iterative method. Numerical experiments on distorted meshes confirm second-order spatial accuracy and robust IRP preservation, outperforming a nine-point scheme that fails to maintain the invariant region.

Abstract

We present a finite volume method preserving the invariant region property (IRP) for the reaction-diffusion systems with quasimonotone functions, including nondecreasing, decreasing, and mixed quasimonotone systems. The diffusion terms and time derivatives are discretized by a finite volume method satisfying the discrete maximum principle (DMP) and the backward Euler method, respectively. The discretization leads to an implicit and nonlinear scheme, and it is proved to preserve the invariant region property unconditionally. We construct an iterative algorithm and prove the invariant region property ar each iteration step. Numerical examples are shown to confirm the accuracy and invariant region property of our scheme.
Paper Structure (10 sections, 5 theorems, 71 equations, 15 figures, 7 tables, 1 algorithm)

This paper contains 10 sections, 5 theorems, 71 equations, 15 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Invariant region theory of the model problem
  3. The IRP-preserving finite volume scheme
  4. The iterative method preserving the IRP
  5. Numerical experiments
  6. Example 1
  7. Example 2
  8. Example 3
  9. Example 4
  10. Conclusion

Key Result

Lemma 2.1

Suppose $\kappa_1, \kappa_2 \in [L^{\infty}(Q_T)]^{2\times2}$ are uniformly positive definite in $Q_T$, $(u_0, v_0) \in [H^1(\Omega)]^2$ and there exists $(G_1, G_2) \in\left[W^{2,1}\left(Q_T\right)\right]^2$ such that $(G_1, G_2)|_{S_T}=(g_1, g_2)$. Denote $\Sigma=$$\left[m_1, M_1\right] \times\lef then the coupled system problem_1-problem_2 has a weak solution $(u,v) \in \Sigma$ in $[W_2^{1,1}(Q

Figures (15)

  • Figure 1: The notations
  • Figure 2: The random quadrilateral meshes.
  • Figure 3: The random triangular meshes.
  • Figure 4: The numerical solution $U$ of the IRP-preserving scheme for Example 3 on the random quadrilateral meshes ($U_{\min} = 0$, $U_{\max} = 1$).
  • Figure 5: The numerical solution $V$ of the IRP-preserving scheme for Example 3 on the random quadrilateral meshes ($V_{\min} = 1$, $V_{\max} = 1$).
  • ...and 10 more figures

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Remark 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 2 more