Second Order Runge-Kutta Method (RK2) to Solve ODE

By Kardi Teknomo, PhD.

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Suppose we have ODE then

Formula:

Where

and

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

Example:

, 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. Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\ tutorial\ODE\