The "sixth-jumping" technique came to me having read Adolf Fick's original 1855 (?) scientific paper. It is wise. He knows most "assumptions" on the way to formulating his "Fick's Laws" are not going to be so in most realities,
(I believe you can see "Fickian Diffusion" if you use a radioactive tracer isotope on one side of a boundary (same element; difference in the nucleus not affecting chemical properties), and see it mix in time and have a way to detect concentration of "origin-1" to "origin-2" atoms by radioactivity) - other than that - no chance...)
I saw that if you have "an automatic computer" ("a computer") you don't need to formulate differential equations.
A person of Middle-Eastern origin showed me the computational method for solving mathematical integration ("calculus") approximately but achievably. But having seen that, my "sixth-jumping model" came to me. My sixth-jumping model used as a general solution does have "convergence" with increasing discretisation, by the way, stating the obvious.
The algorithm is / was very efficient. The Computer Science people were very glad of seeing the real performance of computers revealed, by reason of knowing exactly how many operations my algorithm had to do to go each step of the solution.
By the way - when I did my Doctoral research back up to the late
1990's, it really wasn't then possible to solve in 3 dimensions for mathematical expressions for conductive heat flow and diffusion. The computer memory requirement; the computing time. Now; yes - "even I" solve for stresses and strains in 3 dimensions with Finite Element Analysis programs. But then, being realistic... I had 80MB of memory, which was five times a good-spec computer then, and people used to sit there drooling watching the computer go through its boot-up routine and check the memory. But I had to fit a 3-dimensional computational model into that. I did not need or use "swap-space" on a hard-disk - the entire solution fitted into the computer memory.That is a digression from hydrogen in metals.
*** My solution did the right thing. *** That must be surely correct because it explains so much. You talk of "boundaries" and "boundary conditions". How could anyone have prior knowledge of what to set this at??? I found scientific "papers" where solutions were presented which were mathematically correct but physically incorrect. My solution is a model which done sytematically gives a quantitative result. It "shows the way" because it is a model.You talk about "hacks" - but this solution, which is the implementation of a model, is "pure" - you know what it represents physically.
I don't think I am contributing anything, because your comment is very insightful and I get the impression you are a genuine scientist.
I would commend anyone interested to go back and re-read what you have written after reading these comments of mine, because of the quality of your understanding.
PS - I used the "sixth-jumping" solution for two years before I ever explained to anyone how it worked. It took me something like 20 minutes to find an explanation which worked, to a person who was an expert in diffusion and had made useful discoveries. He finally "got it" and asked "So if Adolf Fick had had an automatic computer, he would have solved for diffusion this way?", to which I replied something like "Almost certainly".