各向异性扩散中的保单调性
Preserving Monotonicity in Anisotropic Diffusion
摘要：
我们表明，标准算法的各向异性扩散中心差分（包括最近的对称算法）的基础上不保持单调性。在各向异性热传导的背景下，这可能导致违反热力学第二定律的熵约束，导致热量从较低温度的区域流向较高温度。在温度变化较大的区域，这可能导致温度变为负值。测试案例来说明这一点为中心的非对称和对称差分。提出了基于斜率限制器的算法来解决这些问题，类似于双曲方程二阶格式中使用的算法。虽然中心算法可能适用于许多情况，但有限方法的主要优点是它们保证在存在大温度梯度的情况下避免负温度（这可能导致数值不稳定性）。特别是，有限的方法将有助于模拟热的，稀释的天体物理等离子体，其中传导是各向异性的，温度梯度是巨大的，例如，无碰撞激波和盘-日冕界面。
英文摘要：
We show that standard algorithms for anisotropic diffusion based on centered differencing (including the recent symmetric algorithm) do not preserve monotonicity. In the context of anisotropic thermal conduction, this can lead to the violation of the entropy constraints of the second law of thermodynamics, causing heat to flow from regions of lower temperature to higher temperature. In regions of large temperature variations, this can cause the temperature to become negative. Test cases to illustrate this for centered asymmetric and symmetric differencing are presented. Algorithms based on slope limiters, analogous to those used in second order schemes for hyperbolic equations, are proposed to fix these problems. While centered algorithms may be good for many cases, the main advantage of limited methods is that they are guaranteed to avoid negative temperature (which can cause numerical instabilities) in the presence of large temperature gradients. In particular, limited methods will be useful to simulate hot, dilute astrophysical plasmas where conduction is anisotropic and the temperature gradients are enormous, e.g., collisionless shocks and disk-corona interface.
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