INCONSISTENT INITIAL CONDITIONS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 89

circuits simply because it is generally difficult to obtain the
response of these circuits. Nonlinear circuits are most often analyzed
using LMS-PC algorithms in the time domain, among which LSS-PC
algorithm, i.e. forward Euler as the predictor and backward Euler as
the corrector, is widely used. Although LMS-PC algorithms require the
continuity of network variables, it was demonstrated in [48] that the
backward Euler

is capable of handling Dirac impulses, subsequently inconsistent initial
conditions encountered at switching instants. To illustrate this, consider
step function It represents the discontinuity and its first-order
derivative gives the Dirac impulse function

Using the backward Euler, the derivatives of unit step function
is obtained from

Note that the first-order derivative was obtained from

Eq.(5.14) reveals that the derivatives of can be computed cor-
rectly using backward Euler despite of the discontinuity at
Applying backward Euler to the linear circuit depicted by

where G and C are the conductance and capacitance matrices respec-
tively and

is the input vector, we obtain