2.6 The Runga-Kutta Method

 

1. The improved Euler's method can also be called Runge-Kutta method of order two.

 

2. Runge-Kutta Method of Order Two: The solution of the initial-value problem

 

 

is approximated at the sequence of points (n = 1, 2, 3, 4, ), where is the approximate value of by computing at each step

, (n = 1, 2, 3, ),

with

 

and with h is the selected step size.

 

3. Considering a weighted average of slopes will give an even better approximation for values of the solution. This is the idea for the most frequently used approximation method, the Runge-Kutta method of order four.

 

4. Basic Idea: At each step, the Runge-Kutta method of order four, uses four slopes, one calculated at the left-hand endpoint of the step, one calculated at the right-hand endpoint of the step and two calculated at the midpoint of the step. The two slopes calculated at the midpoint of each step are given twice as much weight as the slopes at the endpoints.

 

5. Runge-Kutta Method of Order Four: The solution of the initial-value problem

 

 

is approximated at the sequence of points (n = 1, 2, 3, ), where is the approximate value of by computing at each step

, (n = 1, 2, 3, ),

 

where . , ,

 

and with h is the selected step size.

 

6. The maximum cumulative error on a bounded interval [a, b] with is

 

 

where C depends on the function and on the interval [a, b] but not on h. Thus, decreasing h greatly decreases the error.

 

 

7. Critical Dependence on Initial Conditions: If slight variations in initial conditions causes large variations in solution curves, then the results of any of the approximation techniques studied here can be misleading.