Lecture Notes in Differential Equations - Bruce E. Shapiro

365
Figure 33.3: Illustration of the Backward’s Euler Method. Instead of constructing
a tangent line with slope f(t 0 , y 0 ) through (t 0 , y 0 ) a line with slope
f(t 1 , y 1 ) is constructed. This necessitates knowing the solution at the t 1 in
order to determine y 1 (t 1 )
y 2
y 1
y 0
t 0 t 1 t 2
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
y = k + hf(t, y) (33.37)
where k = y n−1 . The technique is summarized here:
• Make a first guess at y n and use that in right hand side of 33.36. A
common first guess that works reasonably well is
y (0)
n = y n−1 (33.38)
• Use the better estimate of y n produced by 33.36 and then evaluate
33.36 again to get a third guess, e.g.,
y (ν+1)
n
= y n−1 + hf(t n , y (ν)
n ) (33.39)
• Repeat the process until the difference between two successive guesses
is smaller than the desired tolerance.
It turns out that Fixed Point iteration will only converge if there is some
number K < 1 such that |∂g/∂y| < K where g(t, y) = k + hf(t, y). A more
stable method technique is Newton’s method.