The Solution of Differential Equations on Short-Word-Length Computing Devices

When the numerical solution of differential equations (DE\´s) for dynamical simulation is performed with short-word-length computing devices, it is essential that truncation (round-off) error be understood and controlled where possible. This paper presents a simple but enlightening analysis of truncation errors in the computer solution of DE\´s, an analysis which develops the rationale for the residue register, the distinguishing feature of the digital differential analyzer (DDA), for the suppression of (second-order) truncation error. The analysis is extended formally to complete the theory of nth order truncation error suppression in linear DE\´s. After discussing the application of residue retention techniques to higher order integration formulas, the utility of third-order truncation error suppression strategy is illustrated in the suppression of "limit cycle" oscillations that occur in marginally damped oscillatory systems. The use of residue retention is then shown to be more critical in the case of DE\´s with multiplicative nonlinearities. Finally, design strategies are discussed briefly.