NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 51

Chapter 4
NUMERICAL INTEGRATION OF
DIFFERENTIAL EQUATIONS
The behavior of mixed-mode switching circuits is depicted in the time
domain using differential equations. This chapter is concerned with nu-
merical integration algorithms for differential equations. The conven-
tional linear single-step predictor-corrector (LSS-PC) algorithms and lin-
ear multi-step predictor-corrector (LMS-PC) algorithms are reviewed in
Sections 1 and 2. We show that although these algorithms are robust in
solving both linear and nonlinear circuits, the accuracy of these methods
is limited by the order of polynomials used in extrapolation and the use of
the first-order derivatives. In Section 3, we show that numerical Laplace
inversion that derives the time domain solution from its coun-
terpart is an elegant high-order numerical integration methods for linear
circuits. The accuracy of this method is orders of magnitude higher as
compared with that of LMS-PC algorithms. In addition, this method is
capable of handling both impulses and discontinuities in network vari-
ables that can not be handled by LMS-PC algorithms. This unique
characteristic makes numerical Laplace inversion algorithm particularly
attractive for analysis of mixed-mode switching circuits where impulses
and discontinuities might be encountered at switching instants. We ex-
amine the properties of numerical Laplace inversion by first introducing
Pad é polynomials. Our focus is then shifted to the use of Pad é approxi-
mation in numerical Laplace inversion. The dependence of the accuracy
of numerical Laplace inversion on the time displacement from the time