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An improvement to the Forward Euler method is Heun’s method, a “predictor-corrector” approach that uses the...

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Advance Math

An improvement to the Forward Euler method is Heun’s method, a“predictor-corrector” approach that uses the predicted values atthe next time step to create a second improved (“corrected”)approximation.

Given the first order ODE y? =f?(x,?y), and a point on the solutioncurve (xn?,? yn), we wantto estimate the next point at a step size h later. We makea first prediction of the next value, y*, using theforward Euler approach:

xn+1 ?=? xn +h
y*n+1 ?=? yn +h?f?(xn?,?yn)
But then this predicted value is used to improve our estimate ofthe slope f?. We compute the approximate slope at the endof our timestep as

f?(xn+1,?y*n+1).
We then go back and produce our better estimate ofyn+1 using the average of the twoslopes. i.e.,

yn+1 ?=? yn +h?*(1/2)*[?f?(xn?,?yn)+f?(xn+1,?y*n+1)].
This is Heun's method. It is also identified as one version of thepopular and powerful Runge Kutta methods.

Let y(x) be the solution to the following initialvalue problem,

y? ?=? (3/10)*[?y +sin(xy)],    y?(0) ?=? 1

Use Heun's method with h = .001 to estimate the value ofy(5).

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