Suppose we have ODE then
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
Preferable reference for this tutorial is
Teknomo, Kardi (2015) Solving Ordinary Differential Equation (ODE). http:\\people.revoledu.com\kardi\ tutorial\ODE\