Tuesday, 16 January 2007: 8:45 AM
Non-linear Computational Instability - its Discovery and Initial Solutions. Early Triumphs for Numerical Meteorology and Professor Arakawa
217C (Henry B. Gonzalez Convention Center)
Computational instability was an early problem for numerical weather prediction and numerical simulation of fluid dynamics problems. Although the linear aspects were recognized and largely solved by mathematicians, the early efforts at numerical solution of time-dependent equations with variable advective velocity, mostly by meteorologists, were plagued by unknown sources of instability, generally thought to be non-linear but not well understood. In 1959, Norman Phillips was evidently the first to clearly point out the problem and solve it, to a certain extent, by filtering out the high wavenumber modes that caused it. In those days, before the discovery of fast Fourier transforms, that was not considered a very practical solution, so mostly we just applied large viscous damping and watched everything turn to mush. In 1963, Professor Arakawa began to show, in conference papers and seminars, a technique that he was developing to apply advective algorithms that conserved kinetic energy or enstrophy or both, and thereby automatically eliminated instability associated with advection. Some of us recognized the great merit of such schemes, and I began applying them to convective simulations before Arakawa had published his work, since he was always careful not to rush into print before he was sure he had done the best he could. After his 1966 publication on the subject, in the first issue of the Journal of Computational Physics, it was in the public domain and Akio began receiving the acclaim he deserved. For the next 20 years or so, the Arakawa schemes became the foundation for most successful two- and three-dimensional predictions and simulations. Because most of the work description resided in the meteorological literature, it was not immediately recognized by other fluid mechanists who were starting to apply numerical simulation to their problems. One still sees rediscovery of the Arakawa approach now and then, or at least I did 10 years ago. A problem with the Arakawa methods is that, though stable, they are not more accurate in phase simulation, sometimes not as accurate, as other methods of the same nominal accuracy. In recent years, they have been often replaced by higher order monotonicity schemes, which are usually slightly damped or slightly unstable but have better phase accuracy. Nevertheless, an Arakawa difference scheme is still likely to be the first choice for a grad student or other beginner who wants a method that is almost guaranteed to work.