Numerical analysis [of PDEs]

To find numerical solutions to PDEs on a digital computer one has no choice but to make approximations. In the typical case of the finite difference method one sets up a system with discrete cells in space and time that is much like a continuous cellular automaton, and then hopes that when the cells in this system are made small enough its behavior will be close to that of the continuous PDE.

Several things can go wrong, however. The pictures below show as one example what happens with the diffusion equation when the cells have size dt in time and dx in space. So long as the so-called Courant condition dt/dx < 1/2 is satisfied, the results are correct. But when dt/dx is made larger, an instability develops, and the discrete approximation yields completely different results from the continuous PDE.

Many methods beyond finite differences have been invented over the past 30 years for finding numerical solutions to PDEs. All however ultimately involve discretization, and can suffer from difficulties that are similar—though often more insidious—to those for finite differences.

For equations where one can come at least close to having explicit algebraic formulas for solutions, it has often been possible to prove that a certain discretization procedure will yield correct results. But when the form of the true solution is more complicated, such proofs are typically impossible.

And indeed in practice it is often difficult to tell whether complexity that is seen is actually a consequence of the underlying PDE, or is instead merely a reflection of the discretization procedure. I strongly suspect that many equations, particularly in fluid dynamics, that have been studied over the past few decades exhibit highly complex behavior. But in most publications such behavior is never shown, presumably because the authors are not sure whether the behavior is a genuine consequence of the equations they are studying.