To derive the error, again, we simply use Taylor series, but in an appropriate form. Instead of approximating for y(t + h) given y(t), we approximate for y(t) given y(t + h):
Note that the second term is negative because t = (t + h) − h.
From the differential equation, we may substitute y(1)(t) = f(t, y(t)):
If we substitute y(t) = y0, y(t + y) = y1, and t + h = t1 into this equation, we get:
If we remove the h2 term, we get the equation which we are solving for the backward-Euler's method. With further manipulation, it can be argued that this implies that the error also drops with respect to h2.
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.