Topic 14.6: Stiff Differential Equations (Error Analysis)

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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.