Civil Engineering Reference
In-Depth Information
This is referred to as the
Forward Euler
numerical integration formula‚ and it
is merely a natural consequence of the finite difference approximation‚
where the left hand side represents the derivative evaluated at time
Another alternative is to instead let the left hand side of Equation (2.39)
correspond to the derivative at time
this leads to the numerical integration
formula‚
This technique‚ called the
Backward Euler
method may be interpreted as shown
in Figure 2.5(b)‚ where we take a step back from the time point and use the
derivative at that point to estimate the value there. Of course‚ the value of the
derivative and the value are both unknown‚ so that we obtain an implicit
equation in here‚ instead of the explicit equation provided by the Forward
Euler method‚ where all terms on the right hand side are known. It can be seen
then that both Equations (2.38) and (2.40) effectively convert the differential
equation to a nonlinear equation at each time point.
The intuition behind yet another formula for numerical integration can like-
wise be arrived at. The Trapezoidal rule follows similar principles as the For-
ward Euler and Backward Euler methods‚ but averages the derivative at the
points
and
so that it may be stated as
As a motivating example to see how these techniques could be applied to
circuit analysis‚ let us consider a linear capacitor of value
C
that is represented
by the device equation
where and are, respectively, the current through and voltage across the
capacitor, and is the time variable
5
. Let us assume that we know the value
of this voltage and current at some time point
and that we would like to
calculate its value at the time point
for a sufficiently small time
step,
Using the Forward Euler method, we have
