The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 4. IMPROVING ON EULER’S METHOD 81
Thus the region of absolute stability for the Backwards Euler Method encompasses
the entire complex plane outside the circle of radius 1 centered at the point (1, 0),
as illustrated in figure 4.6.
Figure 4.6: The gray area indicates the region of absolute stability for the Backwards
Euler Method.
The Backwards Euler Method is an example of an implicit method, because
it contains y n implicitly. In general it is not possible to solve for y n explicitly as a
function of y n−1 in equation 4.98, even though it is sometimes possible to do so for
specific differential equations. Thus at each mesh point one needs to make some first
guess to the value of y n and then perform some additional refinement to improve
the calculation of y n before moving on to the next mesh point. A common method
is to use fixed point iteration on the equation
where k = y n−1 . The technique is summarized here:
y = k + hf(t, y) (4.103)
• Make a first guess at y n and use that in right hand side of 4.98. A common
first guess that works reasonably well is
y (0)
n = y n−1 (4.104)
• Use the better estimate of y n produced by 4.98 and then evaluate 4.98 again
to get a third guess, e.g.,
y (ν+1)
n
= y n−1 + hf(t n , y (ν)
n ) (4.105)
• Repeat the process until the difference between two successive guesses is
smaller than the desired tolerance.
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge