I remember my introduction to difference equations. Sometime in 1950 Professor Bohnenblust spoke at an afternoon seminar at Cal-Tech on “difference equations” which were evidently new to most of the audience. I did not attend Cal-Tech but they welcomed outsiders. Bohnenblust contrasted difference equations with differential equations and presented them as a technique for arguing the existence of solutions to the differential equations and sometimes providing information about those solutions. Computers were not mentioned but it was clear to all that these results bore on the approximate solution of differential equations by difference equations with computers. Indeed that was the electricity in the air. The emphasis was on the question of how the computable solution to difference equations approached the solution to the corresponding differential equations. I remember that there were conditions on the size of Δt. I suppose that they were those later described in von Neumann and Richtmyer’s book on such things as numerical stability.
These issues were the main problem in Richardson’s approach. Richardson had foreseen difference equations but not the limitations on Δt. In short difference equations can carry signals only at mesh velocity, Δx/Δt. If this velocity is slower than the signals being modeled, the numbers will not only be badly wrong, but will go to infinity. For the weather the equations model sound, even though sound is not of interest. The calculations blow up trying to model imaginary sound waves that must travel faster than any mesh velocity that Richardson could afford without computers.