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<p>In this paper, we develop and analyze a hybrid high-order (HHO) numerical scheme for the FitzHugh-Nagumo (FHN) reaction-diffusion model which arises in neuronal dynamics, cardiac physiology, and cell division. The proposed method combines HHO for spatial discretization with the BDF2 scheme for time integration, achieving higher-order accuracy and stability. The method supports an arbitrary order of approximation and general polytopal meshes, making it highly flexible. The key ingredients involve local reconstruction and high-order stabilization terms, ensuring robustness. A Lyapunov functional is introduced to establish boundedness, and an elliptic-type projection is used for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-error estimation. Numerical experiments confirm the theoretical results and demonstrate the efficiency of the HHO-BDF2 framework in simulating excitable media models.</p>

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Rigorous HHO Optimal Error Estimates and Simulation of FitzHugh-Nagumo Model

  • 期刊论文

  • 发表日期：

    2025年11月16日

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  • Ajeet Singh,
  • Ram Jiwari

摘要

In this paper, we develop and analyze a hybrid high-order (HHO) numerical scheme for the FitzHugh-Nagumo (FHN) reaction-diffusion model which arises in neuronal dynamics, cardiac physiology, and cell division. The proposed method combines HHO for spatial discretization with the BDF2 scheme for time integration, achieving higher-order accuracy and stability. The method supports an arbitrary order of approximation and general polytopal meshes, making it highly flexible. The key ingredients involve local reconstruction and high-order stabilization terms, ensuring robustness. A Lyapunov functional is introduced to establish boundedness, and an elliptic-type projection is used for \(L^2\) L 2 -error estimation. Numerical experiments confirm the theoretical results and demonstrate the efficiency of the HHO-BDF2 framework in simulating excitable media models.