Improving the analog simulation of partial differential equations by hybrid computation

Abstract

The conventional analog computer simulation of partial differential equations is very useful, especially where fast repetitive runs are desired. However, the rough approxi mations in the space derivatives obtained by differencing one or more of the space variables in a time-independent equation can lead to considerable error. We propose here several methods of correcting this error which require hy brid computation for their meaningful use. The integrator outputs from each "cell" of the analog computer are sampled simultaneously, and these sample values are then processed numerically in the digital com puter to produce corrections to the derivative first-order approximations being mechanized on the analog com puter. These corrections are supplied to the analog solu tion through the DAC as a continuous correction function throughout the time transient. We have, so far, found that the best method of computing these corrections is based on the truncated Fourier series. We also found that axis transformations can be used very simply and effectively to compute better derivative approximations. By the transformation, samples are bunched in the more active and spread in the less active space regions in a type of adaptive sampling. The digital computer easily computes the necessary transformations and automatically sets the analog potentiometers. Using these two methods, together with some calcula tions to prevent error-noise corruption, we obtained error corrections of the order of 15 to 30 percent in some de rivative approximations now in common use.