**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.