The equation governing the streaming of a quantity down its gradientsuperficially looks similar to the simple constant velocity advection equation.In fact, it is the same as an advection equation if there are no local extremain the computational domain or at the boundary. However, in general when thereare local extrema in the computational domain it is a non-trivial nonlinearequation. The standard upwind time evolution with a CFL-limited time stepresults in spurious oscillations at the grid scale. These oscillations, whichoriginate at the extrema, propagate throughout the computational domain and areundamped even at late times. These oscillations arise because of unphysicallylarge fluxes leaving (entering) the maxima (minima) with the standardCFL-limited explicit methods. Regularization of the equation shows that it isdiffusive at the extrema; because of this, an explicit method for theregularized equation with $\Delta t \propto \Delta x^2$ behaves fine. We showthat the implicit methods show stable and converging results with $\Delta t\propto \Delta x$; however, surprisingly, even implicit methods are not stablewith large enough timesteps. In addition to these subtleties in the numericalimplementation, the solutions to the streaming equation are quite novel:non-differentiable solutions emerge from initially smooth profiles; thesolutions show transport over large length scales, e.g., in form of tails. Thefluid model for cosmic rays interacting with a thermal plasma (valid at spacescales much larger than the cosmic ray Larmor radius) is similar to theequation for streaming of a quantity down its gradient, so our method will findapplications in fluid modeling of cosmic rays.

Numerical Implementation of Streaming Down the Gradient: Application to Fluid Modeling of Cosmic Rays and Saturated Conduction