I often solve pdes for my research, and years ago I found pdetoode in this forum is very handy and solve severy chanlenging pdes (here, here and here for example) in this forum. Thus, I decied to figure its working principle.

I have been studying hard on the package pdetoode by xzczd for many times and many days each time.

I still can't figure out the how the pde converted to odes.

In particularlly, the following command:

((u : func) | Derivative[dx1 : pat, dt_, dx2___][(u : func)])[
  x1 : pat, t_, x2___] :>
 (Sow@coordtoindex@{x1, x2};
  fdd[{dx1, dx2}, {grid}, Outer[Derivative[dt][u@##]@t &, grid], 
   "DifferenceOrder" -> o, PeriodicInterpolation -> periodic])

I guess from the context that pat here means whatever repeats itself several times exactly. However, for code here, after I print dx1, and x1. In some cases dx1 = 01, x1=xy , this confues me a lot.

How can this be? Maybe I misunderstand something here? What does this piece of code try to do?

I tried my best but I still fails to understand it.

Can anyone explain something on the above code or pdetoode? Thanks!