> >Sometimes numerical solutions "blow up" because of instability in the > >numerical scheme. (To see this for yourself, discretize the heat > >equation u_t=u_xx, > [snip] > One solution is to use simulation, not computation. This is what I've > been talking about all the time. The final test for an aircraft > design, before the maiden flight, is the wind tunnel - equations alone > won't do. Well, we can set up a simulated wind tunnel inside our > computers, and we don't necessarily need to put in the Navier-Stokes > equations, either in their closed form or in their numerical form.
If you're going to simulate a wind tunnel in your computer, then you are doing computation. Whether the computations you do to simulate your wind tunnel are based on Navier-Stokes or other models is irrelevant; the numerical stability and accuracy must be estimated if you want to use the results.
If you're going to build a prototype and put it into a real wind tunnel, then you're not doing modelling, you're doing prototyping. If you want to build a model rather than a prototype, then you are doing modelling, but you had better understand how things change with the scale, which requires at least a superficial understanding of the mathematical model.
> Many blowups due to numerical instability can be negotiated by > increasing the precision.
You obviously did not investigate the heat equation as I suggested you do. Increasing the precision *cannot* fix the problems caused by numerical instability. These problems are much more fundamental than that. (Blindly subdividing the grid into more squares similarly cannot fix the problems.)
> >This is not directly relevant to a K-12 education discussion, but it > >does show that raw computational power may not be sufficient to obtain > >correct answers from a model. > > If we had orders of magnitude more computational power than we have > today, we could perform the numerical method with an > infinite-precision numerical package written in Lisp:
And if you don't pay attention to the mathematics of numerical instability, your solution will blow up in just as many steps as when you perform the same method with single-precision floats on a 1970s era computer.