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The accuracy of the Euler method improves only linearly with the step size is decreased, whereas the Heun Method improves accuracy quadratically
.<ref>{{cite web|url=https://web.archive.org/web/20181014204120/http://livetoad.org/Courses/Documents/214a/Notes/euler-heun_method.pdf|title=The Euler-Heun Method|last=|first=|date=|website=|publisher=LiveToad.org|url-status=live|archive-url=|archive-date=|access-date=}}</ref> The scheme can be compared with the [[Explicit and implicit methods|implicit]] [[trapezoidal method]], but with <math>f(t_{i+1},y_{i+1})</math> replaced by <math>f(t_{i+1},\tilde{y}_{i+1})</math> in order to make it explicit. <math>\tilde{y}_{i+1}</math> is the result of one step of [[Euler's method]] on the same initial value problem. So, Heun's method is a [[predictor-corrector method]] with forward [[Euler's method]] as predictor and [[trapezoidal method]] as corrector.▼
▲}}</ref> The scheme can be compared with the [[Explicit and implicit methods|implicit]] [[trapezoidal method]], but with <math>f(t_{i+1},y_{i+1})</math> replaced by <math>f(t_{i+1},\tilde{y}_{i+1})</math> in order to make it explicit. <math>\tilde{y}_{i+1}</math> is the result of one step of [[Euler's method]] on the same initial value problem. So, Heun's method is a [[predictor-corrector method]] with forward [[Euler's method]] as predictor and [[trapezoidal method]] as corrector.
==Runge–Kutta method==
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