沿梯度向下流动的数值实现：应用于宇宙射线和饱和传导的流体模拟
Numerical Implementation of Streaming Down the Gradient: Application to Fluid Modeling of Cosmic Rays and Saturated Conduction
摘要：
从表面上看，控制一个量沿其梯度向下流动的方程与简单的常速平流方程相似。实际上，如果在计算区域或边界处没有局部极值，则它与平流方程相同。然而，一般情况下，当计算区域中存在局部极值时，它是一个非平凡的非线性方程。标准的迎风时间演变与CFL有限的时间步长在网格尺度上的寄生振荡的结果。这些振荡，这起源于极值，传播整个计算域，甚至在后期是无阻尼的。这些振荡出现，因为非物理大通量离开（进入）的最大值（最小值）与标准CFL有限的显式方法。方程的正则化表明它在极值处是扩散的;因此，用$Δt \propto Δx^2$正则化方程的显式方法表现良好。我们发现，隐式方法显示稳定和收敛的结果与$Δt \propto Δx$，然而，令人惊讶的是，即使是隐式方法是不稳定的足够大的时间步长。除了数值实现中的这些微妙之处外，流方程的解是相当新颖的：不可微的解出现在最初光滑的轮廓中;解显示了大长度尺度上的传输，例如，以尾巴的形式。宇宙射线与热等离子体相互作用的流体模型（在比宇宙射线拉莫尔半径大得多的空间尺度上有效）类似于一个量沿其梯度流动的方程，因此我们的方法将在宇宙射线的流体建模中找到应用。
英文摘要：
The equation governing the streaming of a quantity down its gradient superficially looks similar to the simple constant velocity advection equation. In fact, it is the same as an advection equation if there are no local extrema in the computational domain or at the boundary. However, in general when there are local extrema in the computational domain it is a non-trivial nonlinear equation. The standard upwind time evolution with a CFL-limited time step results in spurious oscillations at the grid scale. These oscillations, which originate at the extrema, propagate throughout the computational domain and are undamped even at late times. These oscillations arise because of unphysically large fluxes leaving (entering) the maxima (minima) with the standard CFL-limited explicit methods. Regularization of the equation shows that it is diffusive at the extrema; because of this, an explicit method for the regularized equation with $Δt \propto Δx^2$ behaves fine. We show that the implicit methods show stable and converging results with $Δt \propto Δx$; however, surprisingly, even implicit methods are not stable with large enough timesteps. In addition to these subtleties in the numerical implementation, the solutions to the streaming equation are quite novel: non-differentiable solutions emerge from initially smooth profiles; the solutions show transport over large length scales, e.g., in form of tails. The fluid model for cosmic rays interacting with a thermal plasma (valid at space scales much larger than the cosmic ray Larmor radius) is similar to the equation for streaming of a quantity down its gradient, so our method will find applications in fluid modeling of cosmic rays.
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