By Kardi Teknomo, PhD .

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Suppose we have ODE then

Formula :



Other variation name : Improve Euler method, Heun's method, Midpoint method

Error term : , correct up to the second order term in Taylor series expansion.

The Taylor series expansion is

The computation is using spreadsheet that can be downloaded here


, set , with initial condition . The few first results and the graph of solution are given below.

Note that after , the solution is not correct. See Comparison.

Expanding the order Runge-Kutta formula, we have

Second term of the right hand side [ ] is the estimated range difference. The estimated range difference is computed based on a half of the slope at the beginning of the interval and the slope in the middle of the interval , as illustrated in the figure below


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See also: Numerical Excel tutorial , Dynamical System tutorial , Kardi Teknomo's Tutorial

This tutorial is copyrighted .

Preferable reference for this tutorial is

Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\\kardi\ tutorial\ODE\