By Kardi Teknomo, PhD .

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

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Runge Kutta-2

Suppose we have ODE Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 then

Formula : Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

Where

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 and Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

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

Error term : Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 , correct up to the second order term in Taylor series expansion.

The Taylor series expansion is Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

The computation is using spreadsheet that can be downloaded here

Example:

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 , set Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 , with initial condition Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 . The few first results and the graph of solution are given below.

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

Note that after Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 , the solution is not correct. See Comparison.

Expanding the Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 order Runge-Kutta formula, we have

Solving Ordinary Differential Equation (ODE) using Runge-Kutta2

Second term of the right hand side [ Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 ] is the estimated range difference. The estimated range difference is computed based on a half of the slope at the beginning of the interval Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 and the slope in the middle of the interval Solving Ordinary Differential Equation (ODE) using Runge-Kutta2 , as illustrated in the figure below

RK2

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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:\\people.revoledu.com\kardi\ tutorial\ODE\