Limitations due to Noise, Stability and Component Tolerance on the Solution of Partial Differential Equations by Differential Analysers†

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Abstract
The solution of partial differential equations by a differential analyser is considered with regard to the effects of noise, computational instability and the deviation of components from their ideal values. It is shown that the ‘serial’ method of solving parabolic, hyperbolic and elliptic equations leads to serious instability which increases as the finite difference interval is reduced. The truncation error (due to the difference approximations) decreases as the interval is made smaller and consequently an ‘optimal’ accuracy is reached when the unstable noise errors match the truncation errors. Evaluation shows that the attainable accuracy is severely limited especially for hyperbolic and elliptic equations. The ‘parallel’ method is stable when applied to parabolic and hyperbolic (but not elliptic) equations and the attainable accuracy is then limited by the accumulation of component tolerances. Quantitative investigation shows how reasonably high accuracy can be achieved with a minimum of precise adjustments.

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