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<p>This paper presents the implementation of an eighth-order finite difference Hermite interpolation based Weighted Essentially Non-Oscillatory (HWENO) scheme combined with the integrating factor Runge-Kutta (IFRK) method to address the nonlinear degenerate parabolic equation. While many existing works use WENO schemes to solve this type of equation, the stencils required for high order accuracy are very wide, leading to reduction on computational efficiency and numerical resolution. To overcome these issues, we propose HWENO scheme to the equation by combining the original equation with its spatial derivative equation. In our previous work[<CitationRef CitationID="CR4">4</CitationRef>, <CitationRef CitationID="CR5">5</CitationRef>], we encountered challenges related to overly stringent Courant-Friedrichs-Lewy (CFL) conditions and limited accuracy. In this paper, we select appropriate stencils and fully utilize derivative information to achieve the optimal order of accuracy for the HWENO reconstruction. Additionally, we introduce a linear diffusion term in the equations, and employ the third-order Strong Stability-Preserving (SSP) IFRK method to relax the CFL condition. As a result, the parabolic time step restriction is removed. Finally, a series of numerical experiments demonstrate the effectiveness of the HWENO-IFRK scheme in terms of high accuracy, high resolution, high efficiency, and robustness.</p>

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Stability-Enhanced HWENO-IFRK Scheme for Parabolic Equation with Nonlinear Degenerate Diffusion

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Muyassar Ahmat,
  • Guoliang Zhang

摘要

This paper presents the implementation of an eighth-order finite difference Hermite interpolation based Weighted Essentially Non-Oscillatory (HWENO) scheme combined with the integrating factor Runge-Kutta (IFRK) method to address the nonlinear degenerate parabolic equation. While many existing works use WENO schemes to solve this type of equation, the stencils required for high order accuracy are very wide, leading to reduction on computational efficiency and numerical resolution. To overcome these issues, we propose HWENO scheme to the equation by combining the original equation with its spatial derivative equation. In our previous work[4, 5], we encountered challenges related to overly stringent Courant-Friedrichs-Lewy (CFL) conditions and limited accuracy. In this paper, we select appropriate stencils and fully utilize derivative information to achieve the optimal order of accuracy for the HWENO reconstruction. Additionally, we introduce a linear diffusion term in the equations, and employ the third-order Strong Stability-Preserving (SSP) IFRK method to relax the CFL condition. As a result, the parabolic time step restriction is removed. Finally, a series of numerical experiments demonstrate the effectiveness of the HWENO-IFRK scheme in terms of high accuracy, high resolution, high efficiency, and robustness.