The second law of thermodynamics is a useful and universal tool to derive the generalizations of the Fourier's law. In many cases, only linear relations are considered between the thermodynamic fluxes and forces, i.e., the conduction coefficients are independent of the temperature. In the present paper, we investigate a particular nonlinearity in which the thermal conductivity depends on the temperature linearly. Also, that assumption is extended to the relaxation time, which appears in the hyperbolic generalization of Fourier's law, namely the Maxwell-Cattaneo-Vernotte (MCV) equation. Although such nonlinearity in the Fourier heat equation is well-known in the literature, its extension onto the MCV equation is rarely applied. Since these nonlinearities have significance from an experimental point of view, an efficient way is needed to solve the system of partial differential equations. In the following, we present a numerical method that is first developed for linear generalized heat equations. The related stability conditions are also discussed.

中文翻译：

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非线性傅里叶和 Maxwell-Cattaneo-Vernotte 热传输方程的数值处理

热力学第二定律是推导出傅立叶定律推广的有用且通用的工具。在许多情况下，只考虑热力学通量和力之间的线性关系，即传导系数与温度无关。在本文中，我们研究了一种特定的非线性，其中热导率线性地取决于温度。此外，该假设还扩展到弛豫时间，它出现在傅立叶定律的双曲线推广中，即 Maxwell-Cattaneo-Vernotte (MCV) 方程。尽管傅立叶热方程中的这种非线性在文献中是众所周知的，但很少将其扩展到 MCV 方程。由于这些非线性从实验的角度来看很重要，需要一种有效的方法来求解偏微分方程组。在下文中，我们提出了一种首先为线性广义热方程开发的数值方法。还讨论了相关的稳定性条件。