Numerical Implementation of Streaming Down the Gradient: Application to
  Fluid Modeling of Cosmic Rays and Saturated Conduction
  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 $\Delta t \propto \Delta x^2$ behaves fine. We show
that the implicit methods show stable and converging results with $\Delta t
\propto \Delta x$; 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.
