Runge-Kutta Method

Runge-Kutta Method: 

The objective of the Runge-Kutta method is that it is used to solve the ordinary differential equation. Small the size of h we get a more accurate solution of the ordinary differential equation.

In this article, we have discussed the Runge-Kutta method that is

• First-order Runge-Kutta method
• Second-order Runge-Kutta method
• Third-order Runge-Kutta method
• Fourth-order Runge-Kutta method

Lastly, we have discussed a few examples depending on the Runge-Kutta method also provides a few examples for practice.

Thus, we can conclude that the most commonly used method for solving the first-order initial value problems is the classical Runge-Kutta method of fourth-order because using two slopes in the method, we can obtain methods of second order, which are called as the second-order Runge-Kutta methods. The method has one arbitrary parameter, whose value is suitably chosen. 

The methods using four evaluations of slopes have two arbitrary parameters. All these methods are of fourth-order, that is the truncation error is of order O(h5). The values of these parameters are chosen such that the method becomes simple for computations. One such choice gives the classical Runge-

Kutta method of fourth-order. If we use five slopes, we do not get a fifth-order method, but only a fourth-order method. It is due to this reason, the classical fourth-order Runge-Kutta method is preferred for computations.