Theorem  (Runge-Kutta Method of order 4)  Assume that  f(t,y)  is continuous and satisfies a Lipschits condition in the variable  y,  and consider the  I. V. P. (initial value problem)

         with ,  over the interval  .
        
The Runge-Kutta method uses the formulas ,  and  

             for    
where
           

as an approximate solution to the differential equation using the discrete set of points  .  

Proof  The Runge-Kutta Method  The Runge-Kutta Method  

Theorem  (Precision  of the Runge-Kutta Method of Order 4)  Assume that    is the solution to the I.V.P.    with  .  If    and    is the sequence of approximations generated by the Runge-Kutta method of order 4, then at each step, the local truncation error is of the order  ,  and the overall global truncation error   is of the order

        ,  for  .  


The error at the right end of the interval is called the final global error  

        .  

Proof  The Runge-Kutta Method  The Runge-Kutta Method  

Animations (Runge-Kutta Method of Order 4  Runge-Kutta Method of Order 4).  Internet hyperlinks to animations.

Algorithm (Runge-Kutta Method). To compute a numerical approximation for the solution of the initial value problem   with    over   at a discrete set of points using the formula  

    ,  for    
where
    ,
    ,
    , and
    .

Computer Programs  The Runge-Kutta Method  The Runge-Kutta Method  

Mathematica Subroutine (Runge-Kutta Method of Order 4).

Example 1.  Solve the I.V.P.  .  
Solution 1.

Example 2.  Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 2.

Example 3.  Plot the error for Runge-Kutta's method.
Solution 3.

Example 4.  Reduce the step size by   and see what happens to the error.
Recalculate points for Runge-Kutta's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Runge-Kutta's method.
Solution 4.

Example 5.   Solve    with    over  .
Solution 5.

Example 6.  Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 6.

Example 7.  Plot the absolute value of the error for Runge-Kutta's method.
Solution 7.

Example 8.  Reduce the step size by   and see what happens to the error.
Recalculate points for Runge-Kutta's method, and the analytic solution using twice as many subintervals.
Then Plot the error for Runge-Kutta's method.
Solution 8.

Example 9.  Solve the I.V.P.  .  
Solution 9.

Example 10. Use Mathematica to find the analytic solution and graph for the I.V.P.  .  
Solution 10.

Various Scenarios and Animations for the Runge-Kutta Method for O.D.E's

Example 11.  Solve the I.V.P.  .    Compute the Runge-Kutta solution to the I.V.P.
Solution 11.

Animations (Runge-Kutta Method of Order 4  Runge-Kutta Method of Order 4).  Internet hyperlinks to animations.

Old Lab Project (Runge Kutta Method of order 4  Runge Kutta Method of order 4).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

The Runge-Kutta Method  The Runge-Kutta Method  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook Runge Kutta Method for O.D.E.'s

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(c) John H. Mathews 2004