Heun's method: Difference between revisions

Revision as of 16:53, 10 August 2012

Revision as of 18:02, 16 September 2012

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==Description==

Euler’s method is used as the foundation for Heun’s method. Where Euler’sEuler's method uses the line tangent line to the actualfunction solutionat the beginning of the curveinterval as an estimate of the curveslope itself, providedof the stepfunction sizeover remainsthe small enoughinterval, contendsassuming that if the twostep willsize notis driftsmall, toothe farerror apartwill be small. In realityHowever, even when extremely small step sizes are used, over a large number of steps the error starts to accumulate and boththe estimate diverges from the actual solutionsfunctional divergevalue.

Where the solution curve is concave up, its tangent line will underestimate the vertical coordinate of the next point and vice versa for a concave down solution. The ideal prediction line would hit the curve at its next predicted point. In reality, there is no way to know whether the solution is concave-up or concave-down, and hence if the next predicted point will overestimate or underestimate its vertical value. The concavity of the curve cannot be guaranteed to remain consistent either and the prediction may overestimate and underestimate at different points in the ___domain of the solution.