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<p>This study investigates the propagation of nonlinear heat waves in a rigid thermal conductor, where classical models do not accurately capture high-frequency thermal behavior. To address this, we propose a compact finite difference method (CFDM) with spectral–like resolution. The model incorporates non-linear thermal conductivity and a temperature-dependent second sound speed. Using a Padé–based compact scheme, we discretize the governing equations and analyze stability through the von Neumann method, confirming unconditional stability. Numerical simulations demonstrate the high accuracy, rapid convergence, and reduced numerical oscillations of the method compared to existing approaches. Furthermore, an error estimation framework based on Sobolev space norms supports the theoretical convergence claims. These results establish CFDM as an efficient and robust tool for modeling non-linear thermal wave phenomena.</p>

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Compact finite difference schemes with spectral-like resolution for solving nonlinear heat-wave propagation in a rigid thermal conductor

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月28日

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  • N. H. Sweilam,
  • N. Bahaa El Din,
  • M. K. Ammar,
  • E. M. Abo-Eldahab

摘要

This study investigates the propagation of nonlinear heat waves in a rigid thermal conductor, where classical models do not accurately capture high-frequency thermal behavior. To address this, we propose a compact finite difference method (CFDM) with spectral–like resolution. The model incorporates non-linear thermal conductivity and a temperature-dependent second sound speed. Using a Padé–based compact scheme, we discretize the governing equations and analyze stability through the von Neumann method, confirming unconditional stability. Numerical simulations demonstrate the high accuracy, rapid convergence, and reduced numerical oscillations of the method compared to existing approaches. Furthermore, an error estimation framework based on Sobolev space norms supports the theoretical convergence claims. These results establish CFDM as an efficient and robust tool for modeling non-linear thermal wave phenomena.