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Euler’s method is used as the foundation for Heun’s method. Where Euler’s method uses the tangent line to the actual solution curve as an estimate of the curve itself, provided the step size remains small enough, contends that the two will not drift too far apart. In reality, even when extremely small step sizes are used, over a large number of steps the error starts to accumulate and both solutions diverge.
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
Heun’s Method addresses this problem by considering the interval spanned by the tangent line segment as a whole. Taking a concave-up example, the left tangent prediction line underestimates the slope of the curve for the entire width of the interval from the current point to the next predicted point. If the tangent line at the right end point is considered (which can be estimated using Euler’s Method), it has the opposite problem
<ref>{{cite web
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|archiveurl=http://web.archive.org/web/20090212005921/http://calculuslab.deltacollege.edu/ODE/7-C-2/7-C-2-h.html
|archivedate=2009-02-12}}</ref>
The points along the tangent line of the left end point have vertical coordinates which all underestimate those that lie on the solution curve, including the right end point of the interval under consideration. The solution is to make the slope greater by some amount. Heun’s Method considers the tangent lines to the solution curve at ''both'' ends of the interval, one which ''overestimates'', and one which ''underestimates'' the ideal vertical coordinates. A prediction line must be constructed based on the right end point tangent’s slope alone, approximated using Euler's Method. If this slope is passed through the left end point of the interval, the result is evidently too steep to be used as an ideal prediction line and overestimates the ideal point. Therefore, the ideal point lies approximately half way between the erroneous
[[File:Heun's Method Diagram.jpg|thumb|right|alt=Heun's Method.|A diagram depicting the use of Heun's method to find a less erroneous prediction when compared to the lower order Euler's Method]]
Euler’s Method is used to roughly estimate the coordinates of the next point in the solution, and with this knowledge, the original estimate is re-predicted or ''corrected'' <ref>
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