5.4 Runge-Kutta Methods
The Taylor methods outlined in the previous section have the desirable
property of high-order local truncation error, but the disadvantage of
requiring the computation and evaluation of the derivatives of f  t , y  .
This is a complicated and time-consuming procedure for most problems, so
the Taylor methods are seldom used in practice.
Runge-Kutta methods have the high-order local truncation error of the
Taylor methods while eliminating the need to compute and evaluate the
derivatives of
f  t , y  . Before presenting the ideas behind their
derivation, we need to state Taylor’s Theorem in two variables. The proof
of this result can be found in any standard book on advanced calculus (see,
for example, [Fu, p.331]).
Theorem 5.13 Suppose that f  x, y  and all its partial derivatives of order
less than or equal to n  1are continuous on D   t , y  | a  t  b, c  y  d  ,
and let  t0 , y0   D . For every
t, y   D ,
there exists  between t and t 0
and  between y and y0 with
f  t , y   Pn t , y   Rn t , y  ,
where


f
f
Pn  t , y   f  t0 , y0    t  t0   t0 , y0    y  y0   t0 , y0  
t
y


2
  t  t0   2 f
2 f

t
,
y

t

t
y

y
 0 0   0 
 t 0 , y0 
0
t 2
t y
 2
2

y  y0   2 f


t , y0   
2  0
2
y

 1 n  n

n f
n j
j
      t  t0   y  y0  n  j j  t0 , y0  
t y
 n ! j 0  j 

and
Rn  t , y  
1 n 1  n  1
 n 1 f
n 1 j
j
t

t
y

y




 ,  .
0
0


j 
t n  j y j
 n  1! 
j 0 
■
The function Pn  t , y  is called the n th Taylor polynomial in two variables
for the function
f about  t0 , y0  , and Rn  t , y  is the remainder term
1
associated with Pn  t , y  .
The first step in deriving a Runge-Kutta method is to determine values
for a1 ,1 , and 1 with the property that a1 f  t  1 , y  1  approximates
T  2  t , y   f  t , y  
h
f ' t, y  ,
2
with error no greater than O  h 2  , the local truncation error for the
Taylor method of order two. Since
f ' t, y  
df
f
f
t, y   t, y   t , y   y ' t 
dt
t
y
and
y ' t   f t , y  ,
this implies
T
2
t, y   f t, y  
h f
h f
t, y  
t, y   f t, y .
2 t
2 y
(5.18)
Expanding f  t  1 , y  1  in its Taylor polynomial of degree on about
t, y 
gives
a1 f  t  1 , y  1   a1 f  t , y   a11
f
t , y 
t
f
 a1 1  t , y   a1  R1  t  1 , y  1  ,
y
(5.19)
where
R1  t  1 , y  1  
12  2 f
2 t 2
 ,    11
12  2 f
2 f

,


 
 ,   , (5.20)
ty
2 y 2
for some  between t and t  1 and  between y and y  1 .
Matching the coefficients of f and its derivatives in Eqs. (5.18) and
(5.19) gives the three equations
f
f  t , y  : a1  1;
t, y  :
t
and
f
t, y  :
y
a11 
h
a11  ;
2
h
f t, y  .
2
The parameters a1 ,1 , and 1 are uniquely determined to be
2
a1  1,
h
2
1  ,
and 1 
h
f t, y  ;
2
so
h
h
 h

 h

T  2  t , y   f  t  , y  f  t , y    R1  t  , y  f  t , y   ,
2
2
 2

 2

and from Eq. (5.20),
2
2
h
h2
2 f
 h
 h  f
R1  t  , y  f  t , y   

,


f
t
,
y




 ,  
2
2
4
t y
 2
 8 t
2
2  f
h2
  f t, y 
 ,   .
8
y 2
If all the second-order partial derivatives of f are bounded, then
h
 h

R1  t  , y  f  t , y  
2
 2

is O  h 2  , the order of the local truncation error of Taylor method of
order two. As a consequence, using the new procedure in place of the Taylor
method of order two might add error, but it does not increase the order
of the error.
The difference-equation method resulting from replacing T 
2
 t , y  in
Taylor’s method of order two by f  t ,   h 2 , y   h 2 f t , y   is a specific
Runge-Kutta method known as the Midpoint method.
Midpoint method:
w0   ,
h
h


wi 1  wi  hf  ti  , wi  f  ti , wi   ,
2
2


for each i  0,1,
, N  1.
Since only three parameters are present in a1 f  t  1 , y  1  and all are
2
needed in the match of T   , we need a more complicated form to satisfy
the condition required for any of the higher-order Taylor methods.
The most appropriate four-parameter form for approximating
T
 3
h
h2
t, y   f t, y   f ' t, y   f "t, y 
2
6
is
a1 f  t , y   a2 f t  2 , y   2 f t , y   ; (5.21)
3
and even with this, there is insufficient flexibility to match the term
2

h 2  f
 t, y  f t , y  ,

6  y

resulting from the expansion of  h 2 6  f "  t , y  . Consequently, the best that
can be obtained from using (5.21) are methods with O  h 2  local truncation
error. The fact that (5.21) has four parameters, however, gives a
flexibility in their choice, so a number of O  h 2  methods can be derived.
One of the most important is the Modified Euler method, which corresponds
1
to choosing a1  a2 
and  2   2  h and has the following
2
difference-equation form.
Modified Euler Method:
w0   ,
h
wi 1  wi   f  ti , wi   f  ti 1 , wi  hf  ti , wi    ,
2
for each i  0,1, 2, , N  1,
The other important O  h 2  method is Heun’s method, which correspond to
1
3
2
a1  , a2  , and  2   2  h , and has the following difference-equation
4
4
3
form.
Heun’s Method:
w0   ,
h
wi 1  wi   f  ti , wi   3 f
4
for each i  0,1, 2,
2
2


 ti  h, wi  hf  ti , wi    ,
3
3


, N  1.
Both are classified as Runge-Kutta methods of order two, the order of their
local truncation error.
EXAMPLE 2 Suppose we apply the Runge-Kutta methods of order two to our
usual example,
y '  y  t 2  1, 0  t  2, y  0  0.5,
with
N  10, h  0.2, ti  0.2i, and w0  0.5 in each case. The difference
equations produced from the various formulas are
4
Midpoint method:
wi 1  1.22wi  0.0088i 2  0.008i  0.218;
Modified Euler method:
wi 1  1.22wi  0.0088i 2  0.008i  0.216;
Heun’s method:
wi 1  1.22wi  0.0088i 2  0.008i  0.2173,
for each i  0,1,
,9 . Table 5.5 lists the results of these calculations.
For this problem, the Midpoint method is superior, followed by Heun’s
method.
■
Table 5.5 (see p277)
Although T 
3
 t , y  can be approximated with error O  h3  by an expression of
the form


f t  1 , y  1 f  t   2 , y   2 f  t , y   ,
involving four parameters, the algebra involved in the determination
of 1 , 1 , 2 , and  2 is quite involved and will not be presented. In fact,
the Runge-Kutta method of order three resulting from this expression is
not generally used. The most common Runge-Kutta method in use is of order
four and, in difference-equation form, is given by the following.
Runge-Kutta method of Order Four:
w0   ,
k1  hf  ti , wi  ,
h
1 

k2  hf  ti  , wi  k1  ,
2
2 

h
1 

k3  hf  ti  , wi  k2  ,
2
2 

k4  hf  ti 1 , wi  k3  ,
wi 1  wi 
for each i  0,1,
1
 k1  2k2  2k3  k4  ,
6
, N  1 . This method has local truncation error O  h 4  ,
provided the solution y  t  has five continuous derivatives. The reason for
introducing the notation k1 , k2 , k3 , k4 into the method is to eliminate the
need for successive nesting in the second variable of f  t , y  (see Exercise
17). Algorithm 5.2 implements the Runge-Kutta method of order four.
5
Runge-Kutta (Oreder Four)
PURPOSE To approximate the solution of the initial-value problem
y '  f  t , y  , a  t  b, y  a    ,
at
 N  1 equally
INPUT
spaced numbers in the interval  a, b :
endpoints a , b ; integer N ; initial condition  ,
OUTPUT
approximation wi to y  ti  at the
Step 1
h  b  a  N ;
 N  1
values of ti .
t0  a ;
w0   ;
OUTPUT
 t0 , w0  .
For i  0,1,
Step 2
Step 3
, N  1 do Steps 3-5.
K1  hf  ti , wi  ;
K2  hf  ti  h 2, wi  K1 2 ;
K3  hf  ti  h 2, wi  K2 2 ;
K4  hf  ti  h, wi  K3  .
Step 4
wi 1  wi   K1  2K2  2K3  K4  6 ; (Compute wi 1 .)
ti 1  a  (i  1)h . (compute ti 1 .)
Step 5
Step6
OUTPUT  ti 1 , wi 1  .
STOP
This produced Runge-Kutta method of order four (Algorithm 5.2) described
in the following subroutine:
SUBROUTINE RK4(N,A,B,ALPHA,T,W)
C******************************************************************
C ALGORITHM 5.2 RUNGE-KUTTA (ORDER 4)
C PURPOSE: TO APPROXIMATE THE SOLUTION TO THE INITIAL VALUE PROBLEM
C
Y' = F(T,Y), A <= T <= B, Y(A) = ALPHA,
6
C
AT (N+1) EQUALLY SPACED NUMBERS IN THE INTERVAL [A,B].
C
C INPUT:
ENDPOINTS A,B; INITIAL CONDITION ALPHA; INTEGER N.
C
C OUTPUT: APPROXIMATION W TO Y AT THE (N+1) VALUES OF T.
C-----------------------------------------------------------------C
INTEGER N
REAL A,B,ALPHA
REAL T(0:10),W(0:10)
REAL XK1,XK2,XK3,XK4,H
INTEGER I
EXTERNAL F
C
STEP 1
H = (B-A)/N
T(0)=A
W(0)=ALPHA
C
STEP 2
DO 110 I=0,N-1
STEP 3
USE XK1,XK2,XK3,XK4 FOR K(1),K(2),K(3),K(4) RESP.
XK1 = H*F(T(I),W(I))
XK2 = H*F(T(I)+H/2,W(I)+XK1/2)
XK3 = H*F(T(I)+H/2,W(I)+XK2/2)
XK4 = H*F(T(I)+H,W(I)+XK3)
STEP 4
COMPUTE W(I+1)
W(I+1) = W(I) + (XK1+2*(XK2+XK3)+XK4)/6
COMPUTE T(I+1)
T(I+1)=A+(I+1)*H
STEP 5
CONTINUE
STEP 6
RETURN
END
C
C
C
C
C
C
110
C
■
EXAMPLE 3 (Section 5.4, p278) Using the Runge-Kutta method of order four
to obtain approximations to the solution of the initial-value problem
y '  y  t 2  1,
0  t  2, y  0  0.5,
with h  0.2, N  10, and ti  0.2i .
7
AL52.f is shown as follows:
C******************************************************************
C Example 3 (Section 5.4) Using RUNGE-KUTTA (ORDER 4) to
C
approximate the solution of the initial-value problem
C
dy/dt=y-t^2+1, y(0)=0.5
C******************************************************************
C
INTEGER N
REAL A,B,ALPHA
REAL T(0:10),W(0:10),Y(0:10)
INTRINSIC EXP,ABS
OPEN(UNIT=10,FILE='AL52.doc',STATUS='UNKNOWN')
C
*** Input initial values
A = 0.0
B = 2.0
ALPHA = 0.5
N = 10
WRITE(10,*) ' T(i)
W(i)
Y(i)
|W(i)-Y(i)|'
WRITE(10,*) '---------------------------------------------'
WRITE(*,*) ' T(i)
W(i)
Y(i)
|W(i)-Y(i)|'
WRITE(*,*) '---------------------------------------------'
CALL RK4(N,A,B,ALPHA,T,W)
DO 10 I = 0,10
C *** Exact solution
Y(I) = (T(I)+1.0)*(T(I)+1.0)-0.5*EXP(T(I))
WRITE(10,99) T(I),W(I),Y(I),ABS(W(I)-Y(I))
WRITE(*,99) T(I),W(I),Y(I),ABS(W(I)-Y(I))
10
CONTINUE
STOP
99
FORMAT(1X,F5.1,3F12.7)
END
REAL FUNCTION F(T,Y)
C====================================================================
C PURPOSE
C
Find the value of function f(t,y) = y-t^2+1
C-------------------------------------------------------------------C
REAL T,Y
F = Y-T*T+1.0
8
RETURN
END
h  0.2, N  10, and ti  0.2i gives the results and errors listed in Table 5.6.
■
Table 5.6
Exact
Runge-Kutta
Order Four
ti
yi  y  ti 
wi
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.5000000
0.8292986
1.2140877
1.6489406
2.1272295
2.6408591
3.1799415
3.7324000
4.2834838
4.8151763
5.3054720
0.5000000
0.8292933
1.2140762
1.6489220
2.1272027
2.6408227
3.1798942
3.7323401
4.2834095
4.8150857
5.3053630
Error
yi  wi
0
0.0000053
0.0000114
0.0000186
0.0000269
0.0000364
0.0000474
0.0000599
0.0000743
0.0000906
0.0001089
The main computation effort in applying the Runge-Kutta methods is the
evaluation of f . In the second-order methods, the local truncation error
is O  h 2  , and the cost is two functional evaluation per step. The
Runge-Kutta method of order four requires 4 evaluations per step, and the
local truncation error is O  h 4  . Butcher (see [But] for a summary) has
established the relationship between the number of evaluations per step
and the order of the local truncation error shown in Table 5.7. This table
indicates why the methods of order less than five with smaller step size
are used in preference to the higher-order methods using a larger step
size.
Table 5.7
Evaluations per 2
3
4
5  n  7 8  n  9 10  n
step
Best
possible
O  h 2  O  h3  O  h 4  O  h n 1 
O  h n  2  O  h n 3 
local
Truncation error
One measure of comparing the lower-order Runge-Kutta methods is described
9
as follows:
The Runge-Kutta method of order four requires four evaluations per step,
so it should give more accurate answers than Euler’s method with
one-fourth the step size if it is to be superior. Similarly, if the
Runge-Kutta method of order four is to be superior to the second-order
Runge-Kutta methods, it should give more accuracy with step size h than
1
a second-order method with step size h ,because the fourth-order method
2
requires twice as many evaluations per step.
An illustration of the superiority of the Runge-Kutta fourth-order method
by this measure is shown in the following example.
EXAMPLE 4 For the problem
y '  y  t 2  1, 0  t  2, y  0  0.5,
Euler’s method with h  0.025 , the Midpoint method with h  0.05 , and the
Runge-Kutta fourth-order method with h  0.1 are compared at the common
mesh points of these methods 0.1, 0.2, 0.3, 0.4, and 0.5. Each of these
techniques requires 20 functional evaluations to determine the values
listed in Table 5.8 to approximate
fourth-order
■
Table 5.8
ti
0.0
0.1
0.2
0.3
0.4
0.5
method
Euler
h  0.025
0.5000000
0.6554982
0.8253385
1.0089334
1.2056345
1.4147264
Exact
0.5000000
0.6574145
0.8292986
1.0150706
1.2140877
1.4256394
y  0.5 . In this example, the
is
Modified
Euler
h  0.05
0.5000000
0.6573085
0.8290778
1.0147254
1.2136079
1.4250141
clearly
superior.
Runge-Kutta
Order Four
h  0.1
0.5000000
0.6574144
0.8292983
1.0150701
1.2140869
1.4256384
Extercise set 5.4
Q4. Use the Modified Euler method to approximation the solutions to each
if the following initial-value problems, give Fortran code and compare
the results to the exact solutions:
(b) y=1+ y t +  y t  , 1  t  3, y 1  0, with h  0.1;
2
Exact solution is y  t   t tan  ln t  .
Q11. Repeat Q4 using the Runge-Kutta method of order four.
10