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To determine the error for Heun's method, we must look at the Taylor series:
Consider the IVP:
y(t0) = y0
We would like to approximate y(t0 + h). The Taylor series gives us that:
Note that we may approximate the 2nd derivative by taking the forward-divided difference formula [i.e., f(1)(x) = (f(x + h) − f(x))/h − ½f(2)(ξ)h] for the derivative of the 1st derivative:
First, let us denote K0 = y(1)(t0) = f(t0, y0) and note that y(t0 + h) ≈ y0 + hK0.
We also observe that y(1)(t0 + h) = f(t0 + h, y(t0 + h)), however, by substituting in our approximation, we have y(1)(t0 + h) ≈ f(t0 + h, y0 + hK0)).
By letting K1 = f(t0 + h, y0 + hK0)), we may rewrite the approximation as
Substituting these into our Taylor series, we get that:
and by canceling terms, we get:
Therefore, the error is O(h3).
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
