A Simplified Derivation and Analysis of Fourth Order Runge Kutta Method

Abstract

The derivation of fourth order Runge-Kutta method involves tedious computation of many unknowns and the detailed step by step derivation and analysis can hardly be found in many literatures. Due to the vital role played by the method in the field of computation and applied science/engineering, we simplify and further reduce the complexity of its derivation and analysis by exploring some possibly well-known works and propose a step by step derivation of the method. We have also shown the stability region graphically

Full text

International Journal of Computer Applications (0975 – 8887)
Volume 9– No.8, November 2010
51
A Simplified Derivation and Analysis of Fourth Order
Runge Kutta Method
Musa H.
Department of Mathematics and
Computer Science
UMYU
Ibrahim Saidu
Faculty of Computer Science and
Information Technology
University Putra,Malaysia.
M. Y. Waziri
Faculty of Science, Department of
Mathematics
University Putra,Malaysia
ABSTRACT
The derivation of fourth order Runge-Kutta method involves
tedious computation of many unknowns and the detailed step by
step derivation and analysis can hardly be found in many
literatures. Due to the vital role played by the method in the
field of computation and applied science/engineering, we
simplify and further reduce the complexity of its derivation and
analysis by exploring some possibly well-known works and
propose a step by step derivation of the method. We have also
shown the stability region graphically
Keywords: Fourth order Runge Kutta Method, Derivation,
Stability Analysis
1. INTRODUCTION
Runge-Kutta formulas are among the oldest and best understood
schemes in numerical analysis. However, despite the evolution of
a vast and comprehensive body of knowledge, it continues to be a
source of active research [7]. Runge-Kutta methods provide a
popular way to solve the initial value problem for a system of
ordinary differential equations [11]:
with a given step length h through the interval ,
successively producing approximations to . We deal
exclusively with the step by step derivation and the stability
analysis of the fourth order Runge-Kutta Method. For a thorough
coverage of the derivation and analysis the reader is referred to
[1,2,3,4,5].
The paper has the following structure: section 2 presents
mathematical formulation and derivation, Section 3 presents the
analysis and section 4 presents the conclusion.
2. MATHEMATICAL FORMULATION
AND DERIVATION
We begin by defining the function as in [1,2,3,4,5 and 6]
1( , , )
nn
y y h x y h


Where
1
1
1
1
1
1
( , , )
( , )
( , ), 2,3,..., 1
s
ii
i
i
i i n ij j
j
i
i ij
j
x y h b k
k f x y
k f x c h y h a k i i
ca








    




1 1 1 2 2 3 3 4 4
1
2 2 21 1
3 3 31 1 32 2
4 4 41 1 42 2 43 3
()
( , )
( , )
( , ( ))
( , ( ))
nn
n
n
n
y y h b k b k b k b k
k f x y
k f x c h y ha k
k f x c h y h a k a k
k f x c h y h a k a k a k
    

  
   
    
The functions are expanded using a Taylor series expansion for
function of two variables. To get the unknowns, we use the
fourth order coefficients of order 4
(1)
1
(2)
1
(3) 2
1
(3)
2
(4) 3
1
(2)
2
(4) 2
3
(4)
4
1
1
2
11
26
1
6
11
6 24
1
8
11
2 24
1
24
i
i
ii
i
ii
i
i ij j
ij
ii
i
i i ij j
ij
i ij j
ij
i ij jk k
ij
b
bc
bc
ba c
bc
b c a c
ba c
ba a c
























Setting the coefficients to zero, we have
1 2 3 4 1b b b b   
(1)

International Journal of Computer Applications (0975 – 8887)
Volume 9– No.8, November 2010
52
2 2 3 3 4 4 1
2
b c b c b c  
(2)
2 2 2
2 2 3 3 4 4 1
3
b c b c b c
(3)
3 32 2 4 42 2 4 43 3 1
6
b a c b a c b a c  
(4)
3 3 3
2 2 3 3 4 4 1
4
b c b c b c  
(5)
3 3 32 2 4 4 42 2 4 4 43 3 1
8
b c a c b c a c b c a c  
(6)
2 2 2
3 32 2 4 42 2 4 43 3 1
12
b a c b a c b a c  
(7)
4 43 32 2 1
24
b a a c 
(8)
We use the simplifying assumptions by Butcher:
1(1 ), 2,3,4
s
i ij i j
iba b c j

  

(9)
Which affect the expression for
(3) (4)
23
,

and
(4)
4

. i.e.
(3) (2) (3)
2 1 1
2
  

(4) (3) (4)
3 1 1
3
  

(4) (2) (3) (4)
4 1 1 2
2
   
  
Now using equation (9) for
2,3 and 4j
we have:
3 32 4 42 2 2
(1 )b a b a b c  
(i)
4 43 3 3
(1 )b a b c
(ii)
44
0 (1 )bc
respectively. (iii)
Now when
4j
in (iii),
41c
and
40b
for a four stage
method.
We substitute
41c
in equations 2, 3 and 5 and solve for
2
b
,
3
b
and
4
b
simultaneously. Therefore equations 2, 3 and 5
becomes
2 2 3 3 4 1
2
b c b c b  
22
2 2 3 3 4 1
3
b c b c b
33
2 2 3 3 4 1
4
b c b c b  
Using crammer’s rule, we first find the determinant of the
coefficient matrix
23
22
2 3 2 3 2 2 3 3
33
23
1
1 ( 1)( )( 1)
1
cc
D c c c c c c c c
cc
     
To solve for
2
b
2
3
23 3 3
3
3
3
11
2( 1)(2 1)
11
3 12
11
4
b
c
c c c
Dc
c
  

23 3 3 3
2 2 3 2 2 3 3 2 2 3 2
( 1)(2 1) 1 2
( 1)( )( 1)
12 12 (1 )( )
b
Dc c c c
b c c c c c c
D c c c c
   
        
To solve for
3
b
3
2
22 2 2
2
3
2
11
2( 1)(2 1)
11
3 12
11
4
b
c
c c c
Dc
c


32 2 2 2
3 2 3 2 2 3 3 3 3 2 3
( 1)(2 1) 1 2
( 1)( )( 1)
12 12 ( )(1 )
b
Dc c c c
b c c c c c c
D c c c c
  
        
To solve for
4
b
4
23
22 2 3 2 3 2 3 2 3
23
33
23
1
2( )(3 4 4 6 )
1
3 12
1
4
b
cc
c c c c c c c c
D c c
cc
    

42 3 2 3 2 3 2 3 2 3 2 3
4 2 3 2 2 3 3 23
( )(3 4 4 6 ) 6 4( ) 3
( 1)( )( 1)
12 12(1 )(1 )
b
Dc c c c c c c c c c c c
b c c c c c c
D c c
       
        
Now to solve for
43
a
, we use equation (ii) i.e. when j=3
Hence, we have
3 3 2 3
2
43 3
4 3 3 2 3 2 3 2 3
(1 ) 12(1 )(1 )
12 (1 )
12 ( )(1 ) 6 4( ) 3
b c c c
c
ac
b c c c c c c c c
  

     
    

International Journal of Computer Applications (0975 – 8887)
Volume 9– No.8, November 2010
53
2 2 3
3 2 3 2 3 3 2
(1 )(2 1)(1 )
( )(6 4( )) 3
c c c
c c c c c c c
  
   
To solve for
32
a
and
42
a
, we use equations (i) (when j=2) and
(8) i.e.
3 32 4 42 2 2
(1 )b a b a b c  
(i)
4 43 32 2 1
24
b a a c 
(8)
From equation (8) above,
2 3 3 2 3 2 3 3 2
32 2 4 43 2 2 3 3 2 2 2 3
12(1 )(1 ) ( )(6 4( ) 3)
1 1 1 1
24 24 6 4( ) 3 (1 )(2 1(1 )
c c c c c c c c c
ac b a c c c c c c c c
     
     
     
3 2 3
22
()
2 (2 1)
c c c
cc


Substituting this value into (i), we have
2 2 3 32
42 4
3 3 2 3 2 3
2
2
2 2 3 2 3 3 3 2 2 2 2 3 2 3
2 3 3 2 3
2 2 3 2 3 2 3
(1 )
1 2 ( ) 12(1 )(1 )
12
(1 )
12 (1 )( ) 12 (1 )( ) 2 (2 1) 6 4( ) 3
(1 ){2(1 )(1 2 ) ( )}
2 ( ){6 4( ) 3)}
b c b a
ab
c c c c c c
c
c
c c c c c c c c c c c c c c
c c c c c
c c c c c c c



   

     

       

    
   
This solution assumes that
2 3 2 3 2 1
0,1, 0,1, , 2
c c c c c   
We choose two free parameters
21
3
c
and
32
3
c
Substituting these values into
4
b
,
3
b
and
2
b
we have:
4
1 2 2 1 4
6 4 3 11
3 3 3 3 38
12 8
12 1 1 3
33
b
    
   
    
    
  
  

  
  
3
11
12 3
33
8
2 2 2 1 8
12 1 9
3 3 3 3
b



  
    

    
    
2
21
12 3
33
8
1 1 1 2 8
12 1 9
3 3 3 3
b
 


  

   

   
   
Using equation (1)
1 2 3 4
1 2 3 4
1
1
3 3 1 1
1 8 8 8 8
b b b b
b b b b
   
    
    
Also
2 21 1
3
ca
Using equation (ii) (when j=3),
4 43 3 3
(1 )b a b c
33
43 4
(1 ) 3 2 8
11
8 3 1
bc
ab

    


Also
2 3 3 2 3
42 2 2 3 2 3 2 3
(1 ){2(1 )(1 2 ) ( )}
2 ( ){6 4( ) 3)}
1 2 2 1 2
1 {2(1 )(1 2 ) ( )}
3 3 3 3 3
1
1 1 2 1 2 1 2
2 ( ){6 4( ) 3)}
3 3 3 3 3 3 3
c c c c c
ac c c c c c c
    
   

     


  
      
Using equation (2) we can obtain
4
c
as
4 4 2 2 3 3
4
1
21 3 1 3 2
2 8 3 8 3 1
1
8
b c b c b c
c
  
   
   
   
   
  
Hence,
4 41 42 43
41 4 42 43 1 ( 1) 1 1
c a a a
a c a a
  
        
Also
3 2 3
32 22
2 1 2 2
()
()3 3 3 9 1
1 1 2
2 (2 1) 2 (2 1)
3 3 9
c c c
acc



   

  
From
3 31 32
c a a
31 3 32 21
1
33
a c a 
     
Finally, we know that
1 11 0ca
.
We have therefore determined all the unknowns in the method
and the method can be written in Butcher’s Tableu [3] as
0 0 0 0 0
1/3 1/3 0 0 0
2/3 -1/3 1 0 0
1/8 3/8 3/8 1/8
Which has the form
1 1 2 3 4
( 3 3 )
8
nn
h
y y k k k k
    

International Journal of Computer Applications (0975 – 8887)
Volume 9– No.8, November 2010
54
1
1
2 2 21 1
3 3 31 1 32 2 1 2
4 4 41 1 42 2 43 3 1 2 3
( , )
( , ) ( , )
33
21
( , ( )) ( , ( )
33
( , ( ) ( , ( )
nn
n n n n
n n n n
n n n n
k f x y
hk
h
k f x c h y ha k f x y
k f x c h y h a k a k f x h y h k k
k x c h y h a k a k a k f x h y h k k k

     
        
         
3. ANALYSIS OF THE METHOD
The stability polynomial is given by
1
( ) 1 ( )
T
R h hb I hA e

  
and it is required that
( ) 1Rh
for absolute stability see [6]. Now for the Runge Kutta forth
order method,
1 1 2 3 4
( 3 3 )
8
nn
h
y y k k k k
    
The Butcher’s Tableu is
0 0 0 0 0
1/3 1/3 0 0 0
2/3 -1/3 1 0 0
1/8 3/8 3/8 1/8
0 0 0 0
10 0 0
311 0 0
3
1 1 1 0
A











,
1 0 0 0
1 0 0
3
10
3
1
h
I hA hh
h h h













,
1 3 3 1 3 3
8 8 8 8 8 8 8 8
Th h h h
hb h 







1
( ) 1 ( )
T
R h hb I hA e

  
1
1 0 0 0
1
1 0 0 1
33 3
11
8 8 8 8 10 1
3
1
h
h h h h
hh
h h h








 
 
 








2
23 2
1 0 0 0
1
1 0 0
31
33
11
8 8 8 8 10
33 1
21
33
h
h h h h hh h
hh
h h h h





















  



2 2 2 3
22
2
32
8 8 8 3 3 8 3 3
1
33 1
8 8 8
11
31
88
8
h h h h h h h h
h
h h h hh
hh
h

   


   
     
   

   




   













2 2 3 2 3 4
2 2 3
2
3 3 2
8 8 24 24 8 24 24 1
33 1
8 8 8 8
11
31
88
8
h h h h h h h
h h h h
hh
h


     





  














2 3 4 2 3 2
33
18 8 24 24 8 4 8 8 8 8
h h h h h h h h h h
          
2 3 4
12 6 24
h h h
h    
For absolute stability
2 3 4
1 1 1
1 1 1
2 6 24
h h h h      
Taking the RHS
2 3 4
1 1 1
11
2 6 24
h h h h    
2 3 4
1 1 1 0
2 6 24
h h h h   
Using Mathematica we get the roots as
NSolve[h+h*h/2+h*h*h/6+h*h*h*h/24==0,h]
{{h-2.78529},{h-0.607353-2.8719 },{h-
0.607353+2.8719 },{h0.}}
We consider 3 cases as it can be found in [1]
Case 1
When

is real and
0


,
The roots are -2.785 and 0

International Journal of Computer Applications (0975 – 8887)
Volume 9– No.8, November 2010
55
Hence the stability interval is
( 2.785,0)h
.
Case 2
2 3 4
23
2
8 8 24 24 1
31
8 4 8
11
31
88
8
h h h h
h h h
hh
h


  




















When
h

is pure and
imaginary,
We set
iy


in the stability polynomial to get
3
24
( ) ( ) ( )
1 ( ) 1
2 6 24
yh yh yh
i yh i    
3
24
( ) ( ) ( )
1 ( ) 1
2 24 6
yh yh yh
iyh i     
Let
t yh
and take the magnitude
2
2 4 3
11
2 24 6
t t t
t
   
     
   
   
2 4 2 4 6 4 6 8 4 4 6
2
11
2 24 2 4 48 24 48 578 6 6 36
t t t t t t t t t t t
t
   
            
   
   
Simplifying, we get
68
11
72 576
tt
  
68
0
72 576
tt
  
Using Mathematica to find the roots we have
NSolve[(-(t^6)/72)+((t^8)/576)0,t]
{{t-
.82843},{t0.},{t0.},{t0.},{t0.},{t0.},{
t0.},{t2.82843}}
The equation is satisfied for
2.82843t
i.e.
22t
Hence the stability interval is
0 2 2h
. i.e.
(0,2 2)h
Case 3 When

is complex with
Re( ) 0


, we set
x iy


in
24
3
( ) ( ) ( )
11
2 6 24
h h h
h
  

    
and plot the boundary of the region by plotting the real and
imaginary parts.
The stability region is plotted using Maple as follows
Re( )h
4. CONCLUSION
In this paper, we have simplified the existing derivation and
analysis of the fourth order Runge-Kutta Method for easy
reference to students and plot the stability region. We also
reduced the complexity of the method by proposing a step by
step derivation approach for better understanding to students.
5. REFERENCES
[1] M.K. Jain, S.R.K. Iyengar, R.K. Jain, (2007), Numerical
Methods for Scientific and Engineering Computing.
[2] J. D. Lambert, (1991), Numerical Methods for Ordinary
Differential Systems, the initial value Problem, John Wiley &
Sons Ltd.
[3] J.C. Butcher, (2003), Numerical Methods for Ordinary
Differential Equations, John Wiley & Sons Ltd.
[4] John R. Dorman, (1996), Numerical Methods for
Differential Equations, a Computational Approach, CRC Press,
Inc.
[5] J. D. Lambert, (1973), Computational Methods in
Ordinary Differential Equations, John Wiley & Sons Ltd
[6] Fudzia Ismail, (2010), Lecture Notes on Numerical
Methods (unpublished), University Putra Malaysia
[7] G. Byrne and Hindmarsh, (1990), RK Methods prove
popular at IMA Conference on Numerical ODE’s,SIAM
News,23/2 pp.14-15.
[8] Lawrence F. Shampine, (1985),Interpolation for Runge-
Kutta Methods.SIAM Journal of numerical
analysis,22/5,pp.1014-1027.