Boundary value technique for initial value
problems based on Adams-type second
derivative methods
S. N. JATOR* and R. K. SAHI
Department of Mathematics,
Austin Peay State University
Clarksville, TN 37044
October 13, 2009
Abstract
In this paper we propose a family of second derivative Adams-type methods (SDAMs) of
order up to 2k + 2 (k is the step number) for initial value problems (IVPs). The methods are
constructed through a continuous approximation of the SDAM which is obtained by multistep
collocation. The continuous approximation is used to obtain initial value methods (IVMs)
which are simultaneously applied to generate all approximations on the entire interval. The
order and the linear stability properties of the methods are discussed. Numerical experiments
are performed and the results compared with those of existing methods in the literature.
AMS Subject Classi…cation : 65L05, 65L06
Key Words: Adams-type, Second derivative, Initial value method, linear stability
1
Introduction
In the past several decades, tremendous attention has been given to the development of numerical
techniques for solving IVPs in ordinary di¤erential equations of the form
y 0 = f (x; y); y(a) = y0 ; x
where f : <
<m ! <m , y; y0
<m , a; b
[a; b]
(1)
<, f satis…es a Lipschitz condition (see Henrici [17]).
Corresponding author. Email: Jators@apsu.edu
1
Several of these techniques are generally of the Runge-Kutta type or linear multistep type (see
Lambert [19], Dahlquist [8], Henrici [17], and Hairer and Wanner [16]). Other methods such as
the hybrid and second derivative methods have also been proposed particularly to overcome the
Dahlquist barrier theorem (see Gear [11], Gragg and Stetter [14], and Butcher [6], Gupta [15], and
Kohfeld and Thompson [18], Enright [9] and Cash [7]).
Most of these methods are implemented in a step-by-step fashion in which on the partition N ,
an approximation is obtained at xn only after an approximation at xn 1 has been computed, where
N
: a = x 0 < x 1 < : : : < xN = b
xn = xn
1
+ h; n = 0; 1; :::; N
h = bNa is the constant step-size of the partition of
index .
N,
N is a positive integer, and n is the grid
A di¤erent approach has been proposed by boundary value methods (BVMs), which discretizes
the problem using linear multistep methods and simultaneously solves the resulting system as
given in Amodio, Golik and Mazzia [1], Brugnano and Trigiante [5], and Ghelardoni and Marzulli
[12]. Boundary value techniques for IVPs were also considered by Axelsson and Verwer [4].
In this paper, we adopt the boundary value technique whereby all approximations (y1 ; y2 ; : : : ; yN )T
(T is the transpose) of the solution of (1) are simultaneously generated on the entire interval. The
advantage of this approach is that the global errors at end of the interval are smaller than those
produce by the step-by-step methods due to the fact that the accumulation of error at each step
is inherent in the step-by-step methods. It is known that BVMs can only be successfully implemented if used together with appropriate auxiliary methods (see Ghelardoni and Marzulli [12]).
In this light, we have proposed basic and auxiliary methods which are obtained from the same
continuous scheme. The continuous SDAM is derived through multistep collocation, see Lie and
Norsett [20], Atkinson [3], Onumanyi et al [21], and Gladwell and Sayers [13]. The continuous
representation generates the basic SDAM and k 1 additional methods which are combined and
used to simultaneously produce approximations yj , for j = 1,...,N to the solution of (1) at points
xj for j = 1,...,N on N . The basic and auxiliary methods are obtained from the same continuous
scheme and are of the same order, hence, possible errors which are due to auxiliary methods of
lower order are avoided as the integration proceeds on the entire interval.
The paper is organized as follows. In section two, we state the forms of the basic and auxiliary
methods as well as the order and local truncation error. In section three, we obtain a continuous
representation F (x) for the exact solution y(x) which is used to generate the main SDAMs and
k-1 additional methods for solving (1). Some particular formulas are given in section four and
their linear analysis discussed in section …ve. Numerical examples are given in section six to show
the e¢ ciency of the methods. Finally, the conclusion of the paper is discussed in section seven.
2
2
Basic and auxiliary methods
In this section, our objective is to derive the basic SDAM and the auxiliary methods.
Basic method. The basic methods are of the form
yn+k
yn+k
1
=h
k
X
2
j fn+j + h
k
X
(2)
j gn+j
j=0
j=0
where j and j are unknown constants. We note that yn+j is the numerical approximation to
the analytical solution y(xn+j ), fn+j = f (xn+j ; y(xn+j )), j = 0; : : : ; k, and
gn+j =
df (x;y(x)) xn+j
jyn+j ,
dx
j = 0; 1; 2; : : : ; k.
The method (2) can be written compactly as
(E)yn = h (E)fn + h2 %(E)gn
where ( ) =
polynomials,
k
(3)
Pk
Pk
j
, and %( ) =
,
( ) =
j=0
j=0 j
C, and E j yn = yn+j is a shift operator.
j
k 1
j
are the characteristic
Auxiliary method. The auxiliary methods are of the form
yn+r
yn+k
1
=h
k
X
2
j fn+j + h
j=0
k
X
j gn+j ;
r = 0; : : : ; k
2
(4)
j=0
Local truncation error and order. Following Fatunla [10] and Lambert [19] we de…ne the
local truncation error associated with (2) to be the linear di¤erence operator
L[y(x); h] =
k
X
j=0
f j y(x + jh)
hy 0
j (x
+ jh)
h2 j y 00 (x + jh)g
(5)
Assuming that y(x) is su¢ ciently di¤erentiable, we can expand the terms in (5) as a Taylor series
about the point x to obtain the expression
L[y(x); h] = C0 y(x) + C1 hy 0 (x) + : : : + Cq hq y q (x) + : : : ;
where the constant coe¢ cients Cq , q = 0; 1; : : : are given as follows:
3
(6)
C0 =
C1 =
C2 =
Pk
Pk
j
j
j=1
j2
j=1
1
2
..
.
Cq =
j
j=0
Pk
1
[
q!
Pk
j=1
jq
Pk
j=0
j
j
j
Pk
j
j
j=1
jq
j=1
q
Pk
Pk
j=0
1
j
q(q
j
1)
Pk
j=1
jq
2
j]
According to Henrici [17], we say that the method (2) has order p if
C0 = C1 = : : : = Cp = 0,
Cp+1 6= 0
therefore, Cp+1 is the error constant (EC) and Cp+1 hp+1 y (p+1) (xn ) the principal local truncation
error at the point xn . The local truncation error (LTE) is given by
LT E = Cp+1 hp+1 y (p+1) (xn ) + O(hp+2 )
3
Continuous approximation
In order to obtain (2) and (4) we proceed by deriving a continuous representation of the SDAM by
seeking to approximate the exact solution y(x) by a continuous approximation F (x) of the form
F (x) =
2k+2
X
`j
(7)
j (x)
j=0
where x 2 [a; b], `j are unknown coe¢ cients and j (x) are polynomial basis functions of degree
2k+2. We then construct a multistep collocation method by letting j (x) = xj , j = 0; 1; : : : ; 2k+2
and imposing the following conditions.
The interpolating function (7) coincides with the analytical solution at the point xn+k
1
The function (7) satis…es the di¤erential equation (1) at the points xn+j ; j = 0; : : : ; k
The second derivative of (7) coincides with the second derivative of the analytical solution
at the points xn+j ; j = 0; : : : ; k
These conditions produce the following set of (2k + 3) equations
4
F (xn+k 1 ) = yn+k
(8)
1
F 0 (xn+j ) = fn+j ; j = 0; 1; 2; : : : ; k
(9)
F 00 (xn+j ) = gn+j ; j = 0; 1; 2; : : : ; k
(10)
which is solved to obtain `j . Our continuous SDAM is constructed by substituting the values of
`j into equation (7). After some manipulation, our continuous approximation is expressed in the
form
F (x) = yn+k
1
+h
k
X
j (x)fn+j
2
+h
j=0
k
X
(11)
j (x)gn+j
j=0
where j (x) and j (x) are continuous coe¢ cients. The continuous method (11) is used to generate
the main SDAM of the form (2) and k 1 auxiliary methods of the form (4) by appropriately
choosing k and j (x) = xj , j = 0; 1; : : : ; 2k + 2.
4
Some particular formulas
A method of order p = 6 and step k = 2. The continuous method (11) is used to generate the
main SDAM of the form (2) and one additional method of the form (4) by choosing j (x) = xj ,
j = 0; 1; : : : ; 6. Thus, evaluating (11) at x = fxn+2 ; xn g, we generate the following main method
and one additional method.
yn+2
yn
yn+1 =
yn+1 =
h2
h
(11fn + 128fn+1 + 101fn+2 ) +
(3gn + 40gn+1
240
240
h
( 101fn
240
128fn+1
11fn+2 ) +
13gn+2 )
h2
( 13gn + 40gn+1 + 3gn+2 )
240
(12)
(13)
A method of order p = 8 and step k = 3 . The continuous method (11) is used to generate
the main SDAM of the form (2) and one additional method of the form (4) by choosing j (x) = xj
, j = 0; 1; : : : ; 8. Thus, evaluating (11) at x = fxn+3 ; xn+1 ; xn g, we generate the following main
method and two additional method.
5
yn+3 yn+2 =
h2
h
(397fn +2403fn+1 +8451fn+2 +6893fn+3 )+
(163gn +2421gn+1 +7659gn+2 1283gn+3 )
18144
30240
(14)
yn+1 yn+2 =
yn
5
yn+2 =
h
h2
( 3fn 109fn+1 109fn+2 3fn+3 )+
( 31gn 1017gn+1 +1017gn+2 +31gn+3 )
224
10080
(15)
h
( 223fn
567
540fn+1
351fn+2
20fn+3 ) +
h2
( 43gn + 144gn+1 + 171gn+2 + 8gn+3 )
945
(16)
Linear stability analysis
The stability analysis is done through linearization in the spirit of Hairer and Wanner [16] where
we consider the usual test equations
y 0 = y;
2
y 00 =
y
which is applied to the form (3) to yield the characteristic equation
k
k 1
k
X
(q
j
+ q2 j )
j
= 0; q = h
(17)
j=0
We note that (17) is a quadratic equation in q and letting = ei we obtain two roots that can
be combined to produce the stability region. In Figure 1, we give the stability regions of the basic
methods ( k = 2 and k = 3). We note that our calculations reveal that the SDAMs have high
order and relatively small error constants as shown in Table 1.
6
Im
4
k=2
2
k=3
Re
8
6
4
2
2
4
Figure 1: Stability Regions for the Basic Methods k = 2; 3
Method
(12)
Order (p)
6
EC (Cp+1 )
(13)
6
1
9450
(14)
8
313
25401600
(15)
8
103
25401600
(16)
8
13
793800
1
9450
Table 1: Order and error constants for SDAMs.
7
6
Numerical examples
Numerical experiments are performed and presented in this section. In particular, a linear system
of dimension 3 and a nonlinear system of dimension 2 are used to demonstrate the e¢ ciency
and accuracy of our technique. All computations were carried out using our written code in
Mathematica 7.0.
Example 1: We consider the following linear IVP considered by Amodio and Mazzia [2] on
the range 0 x 1.
21y1 + 19y2 20y3 ;
y10 =
0
y2 = 19y1 21y2 + 20y3 ;
y30 = 40y1 40y2 + 40y3 ;
y1 (0) = 1
y2 (0) = 0
y3 (0) = 1
The exact solution of the system is given by
y1 (x) = 21 (e 2x + e 40x (cos(40x) + sin(40x)))
y2 (x) = 12 (e 2x e 40x (cos(40x) + sin(40x)))
y3 (x) = 12 (2e 40x (sin(40x) cos(40x)))
This problem was solved using our methods of order p = 6 and order p = 8. The results are
reproduced in Table 2 and compared with the results given in [2]. It is seen from Table 2 that
our method performs better than that in [2]. The accuracy of our method is further explained
by the small values of the error constants displayed in Table 1. The rate of convergence of our
methods are also consistent with the order of the methods. Thus, for this example, our methods
are superior in terms of accuracy. We note that the maximum relative errors displayed in Table 2
are computed as max (jy y(x)j=(1 + jy(x)j)).
Steps
20
40
80
160
320
640
SDAM
k = 2 (p = 6)
Error
2.9
10 3
7.3
10 5
1.8
10 6
3.3
10 8
5.1
10 10
7.7
10 12
Rate
5.3
5.3
5.8
6.0
6.0
Amodio
k = 5 (p = 6)
Error
5.7
10 2
8.7
10 3
4.9
10 4
1.2
10 5
2.2
10 7
3.7
10 9
Rate
2.7
4.2
5.4
5.8
5.9
SDAM
k = 3 (p = 8)
Error
7.5
10 4
1.9
10 5
1.4
10 7
6.4
10 10
2.5
10 12
9.8
10 15
Rate
5.3
7.1
7.7
8.0
8.0
Amodio
k = 7 (p = 8)
Error
2.9
10 2
6.8
10 3
7.8
10 5
4.7
10 7
2.3
10 9
1.3
10 11
Rate
2.1
6.4
7.4
7.7
7.5
Table 2: Relative errors for example 1.
Example 2: We consider the following nonlinear IVP considered by Wu and Xia [22].
8
1002y1 + 1000y22 ;
y10 =
y20 = y1 y2 (1 + y2 );
y1 (0) = 1
y2 (0) = 0
The exact solution of the system is given by
y1 (x) = e
2x
; y2 (x) = e
x
It is obvious from the numerical results in Table 3 that our method (k = 2) performed excellently
for step sizes h = f0:008; 0:006g compared with the method in Wu and Xia [22] where step sizes
h = f0:002; 0:001g were used. Details of the numerical results are given in Table 3.
t
1
10
h
0.008
0.006
SDAM
N
120
1500
y
y1
y2
y1
y2
Error
1.6348
0.0000
2.4815
2.0329
10
14
10
10
24
Table 3: Absolute errors, jy
7
t
1
10
20
h
0.002
0.001
Wu-Xia
N
500
10000
y
y1
y2
y1
y2
Error
2.5606
8.0150
5.5468
6.0936
10
10
10
10
7
8
16
12
y(x)j, ( SDAM, k = 2) for example 2.
Conclusion
A continuous SDAM is proposed and used to obtained the basic and auxiliary methods which
are combined and simultaneously applied to generate approximations to (1) on the entire interval
with the advantage that the global errors at end of the interval are smaller than those due to the
step-by-step methods. This is due to the fact that the accumulation of error at each step inherent
in the step-by-step methods is avoided. The numerical results displayed in Tables 2 and 3 show
that our method is accurate and reliable. Our future research will be focused on applying the
methods to boundary value problems.
9
References
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10
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11