Environmental Engineering Reference
In-Depth Information
origin is studied in detail and a stepping algorithm that can provide
superior numerical accuracy in computing the time domain response of
linear circuits over a time interval of arbitrary length is introduced. The
chapter is summarized in Section 4.
1.
Linear Single-Step
Predictor-Corrector Algorithms
Consider the first-order differential equation
where denotes time and Assume that is con-
tinuously differentiable with respect to time. Let the solution of (4.1)
at time instants
and
be given by
and
respectively.
Expand
in Taylor series at
with the time displacement
If is sufficiently small, the above series can be truncated at the first
order without introducing a large truncation error
Because (4.3) derives from the known point it is known
as the forward Euler formula. The forward Euler formula is explicit in
the sense that it computes directly from the present point whose
value and the first-order derivative are known. Forward Eu-
ler formula is also known as the first-order predictor. Because the trun-
cation error given by where is directly proportional
to (i) the square of the step size and (ii) the second-order derivative of
if varies rapidly with time, the step size must be kept suffi-
ciently small such that a reasonably good prediction of
can be
obtained.
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