Civil Engineering Reference
In-Depth Information
Similar analyses for the Backward Euler method and Trapezoidal rule yield
regions of stability as shown in Figure 2.6(b) and (c)‚ respectively. Several
observations may be made:
When the Backward Euler formula and the Trapezoidal rule are
unconditionally stable for the test equation‚ regardless of the choice of the
step size while the Forward Euler method satisfies this condition only
when is sufficiently small.
When it is the Forward Euler formula and the Trapezoidal rule
that unconditionally obey the requirement that the solution
as
while the Backward Euler method obeys this only when
is
sufficiently small.
The significance of the test equation is that the response of many physical
system can be represented as a sum of exponentials‚ so that stability with
regard to this equation is necessary. Moreover‚ since physical systems tend to
decay with time‚ stability in the left half plane is of greater practical interest‚
and therefore‚ the Backward Euler method and Trapezoidal rule are “good”
methods‚ while the Forward Euler method is “bad” and is often shunned in
practice.
It is important to offer the following caveat: stability is merely a sufficient
condition for goodness‚ and that stability does not imply accuracy. If it did‚
one would need look no further than the Trapezoidal rule for perfection!
2.5.1
Accuracy and the local truncation error
An alternative interpretation of the Forward Euler and Backward Euler meth-
ods views them as truncated Taylor series approximations. Let us consider the
solution at the
time point. Given the solution
at the
time
point‚ we may write the Taylor series approximation of the solution as
We observe from this equation that the Forward Euler method is merely this
approximation‚ truncated after the first order term. Therefore‚ we can explicitly
say that the error for this approximation is given by the truncated terms. In
practice‚ for a “sufficiently small”
6
value of this is dominated by the first
truncated term‚ and we refer to its absolute value as the
local truncation error
(LTE). For the Forward Euler method‚ this is given by
The Backward Euler method can similarly be analyzed‚ this time with the
Taylor series expansion being performed about the point
as
