## Topic 14.6: Stiff Differential Equations (Error Analysis)

Introduction Notes Theory HOWTO Examples Engineering Error Questions Matlab Maple

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.