Preserving monotonicity in anisotropic diffusion
Preserving monotonicity in anisotropic diffusion
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DOI：
10.1016/j.jcp.2007.07.026
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发表时间：
2007-07
期刊：
影响因子：
--
通讯作者：
P. Sharma;G. Hammett
中科院分区：
文献类型：
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作者：
P. Sharma;G. Hammett
我们证明了基于中心差分的各向异性扩散的标准算法（包括最近的对称算法）不保持单调性。在各向异性热传导的情况下，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区域流向较高温度区域。在温度变化较大的地区，这可能导致温度变为负值。给出了中心不对称和对称差分的测试用例来说明这一点。为了解决这些问题，提出了基于斜率限制器的算法，类似于双曲方程的二阶格式。虽然集中算法在许多情况下可能是好的，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）。特别是，有限的方法将有助于模拟热、稀的天体物理等离子体，其中传导是各向异性的，温度梯度很大，例如，无碰撞冲击和磁盘-日冕界面。
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.