Numerical Integration of Differential Equations

Summary

In this chapter, we have reviewed the linear single-step predictor-corrector algorithms and linear multi-step predictor-corrector algorithms, their imitations in analysis of mixed-mode switching circuits. Numerical Laplace inversion that derives the time domain solution from its s-domain response has been introduced. We have shown that Padé approximation based numerical Laplace inversion is a high-order numerical integration method for linear circuits with accuracy orders of magnitude higher as compared with LMS-PC algorithms. In addition, we have shown that this method is capable of handling both impulses and discontinuities in network variables that can not be handled by LMS-PC algorithms. The dependence of the accuracy of numerical Laplace inversion on the time displacement from the time origin has been studied in detail. For a given function, an optimal step size at which the error is the minimal, exists. To improve accuracy, we have shown that the stepping algorithm with the step size set to the optimal step size can provide superior numerical accuracy in computing the time domain response of linear circuits over a time interval of arbitrary length.