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  • $\begingroup$ Thanks so much for the super-detailed answer! I have a couple questions that might be a bit newb-ish: 1. Why is it obvious that the solution isn't dependent on M at low t? 2. I checked the documentation for NDSolve options, but it's not clear to me how I would add more points/increase resolution. I can afford to let it run a bit longer at the moment. $\endgroup$ Commented Jul 28, 2016 at 22:58
  • $\begingroup$ Also I dunno if this is fair to ask, but the reason I wasn't expecting this behavior was that I expected the solution to only increase with time, which makes sense since my initial condition has a negative derivative everywhere, leading to a positive time derivative everywhere. This is clearly not what happens. I attributed this to the oscillations that appeared in the solution near the initial condition; is it possible that this behavior will change with higher resolution? $\endgroup$ Commented Jul 28, 2016 at 23:01
  • $\begingroup$ @basementDweller I did not expect the solution to depend strongly on M, because the coefficient of D[P[M, t], M] is so large, forcing the M-derivative to be exceedingly small to satisfy the PDE. The exception is very near t == 8.15240949872944*^8, where the initial condition forces a rapid variation in M, which must be balanced by a rapid variation in t. So, tiny steps in both M and t are needed to resolve the variations. Use MaxStepSize to set the maximum size of the steps in these two variables. $\endgroup$ Commented Jul 29, 2016 at 0:00
  • $\begingroup$ @basementDweller The oscillations at t == 8.15240949872944*^8 are a symptom, not a cause of the problem at large t. The problem is lack of resolution. Away from that point, the shape of the solution is determined by the shape of the boundary condition at M == 10^5.263. (Remember that information propagates in from the boundary for an advective equation, and the propagation velocity is very high here.) By the way, are you sure that the constants determining the coefficient in the PDE are correct? $\endgroup$ Commented Jul 29, 2016 at 0:10
  • $\begingroup$ the size of the coefficient bothered me for a really long time, and I finally figured out what was wrong. It is indeed missing a factor as written. I'll update it shortly. Changes the behavior significantly too. $\endgroup$ Commented Jul 29, 2016 at 0:24