A new class of a stable implicit schemes for treatment of stiff

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.1, 2013

A New Class of A-stable Implicit Schemes for Treatment of Stiff
System of Ordinary Differential Equations
P.O. Babatola*
Dept. of Mathematical Sciences, Federal University of Technology, Akure, Ondo State, Nigeria
*Email: pobabatola@yahoo.com
Abstract
In this paper, a new class of A-Stable Implicit Rational Runge-Kutta schemes were developed, analyzed and
computerized to solve stiff system of ordinary differential equations. The method is motivated by the Implicit
Conventional Runge – Kutta Schemes and Rational function approximation. While its development and analyses
make use of Taylor series expansion (Taylor and Binomial) and Pade’s approximation techniques respectively. The
schemes are convergent and A-stable.
Keywords: Rational, Runge-Kutta, Consistent, effective, Error bound, Implementation, Convergent, A-stable, A (α)
stable

1. Introduction
An nth order ordinary differential equation is of the general form
y′ = f(x, y), y(xo ) = yo (1.1)
where
yo = (yo, yo2, yo3…yon)
A differential equations (1) whose Jacobian possesses eigen values
λj = Uj + iVj, j = 1(1)n (1.2)
where i = − 1 , satisfying the following conditions.
(a) Ui < < 0, j = 1(1)n
(b) Max U j ( x ) >> min U j (x)
is Stiff. In this case condition (a) show that the system is stable while (b) indicates that the system possesses some
components decay very rapidly.
The problems associated with numerical solution of stiff ODEs were first recognized by Curtis and Hirschfelder
(1952). Other requirement include the necessity for the numerical scheme to be either A-stable, Stiffly stable, A (α)-
stable and A(o) –stable. These stability criteria require that the numerical schemes must be implicit Dahlquist
(1963). In the present of all these problems, Hong Yuanfu (1982) proposed a more general form of this scheme
called Explicit Rational R-K scheme. The general form of the scheme is given by
R
y n + ∑ Wi K i
y n +1 = i =1
R
(3)
1 + y n ∑ Vi H i
i =1

where, K1 = hf(xn, yn)

 S

Ki = hf  x n + ci h, y n + ∑ aij k j  , i = 1(1)R
 i =1 
H1 = hg (xn, zn)

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Vol.3, No.1, 2013

 S

Hi = hg x n + di h, z n + ∑ bij H j  (4)
 i =1 
with g (x n , zn ) = − Z n f (x n, y n )
2
(5)

and Zn = 1 (6)
yn
In his development, aij = 0, bij = 0 for j > i. He developed families of methods of orders two and three of these
schemes. During analysis, he discover that the schemes are A-stable. This property prompted Okunbor (1985) to
develop the order four of this methods. From Okunbor’s work, it is observed that the higher the stage of the method,
the poorer’s the stability. Their performance on stiff oscillatory problem is nothing to write home about.
However, experience with the conventional R-K have shown that Implicit R – K scheme have better resolution
properties than Explicit ones. This expectation is the chief mover of the present consideration by Babatola (1999).

2. The Development of the Proposed Schemes
An R-stage Implicit Rational R-K scheme is of the form
R
y n + ∑ Wi K i
y n +1 = i =1
R
(2.1)
1 + y n ∑ Vi H i
i =1

where,
 i 
Ki = hf  x n + ci h, y n + ∑ aij k j 
 
 j =1 
 i 
Hi = hg  x n + di h, z n + ∑ bij H j  (2.2)
 
 j =1 

1
and g (x n , zn ) = − Z n2 f (x n, y n ) = 2
f ( xn , yn )
yn
with the constraints
R iR
ci = ∑ a ij , d i = ∑ b ij (2.3)
j =1 jj =1

The parameters Vi, Wi, Ci, di, aij and bij are to be determined from the system of non-linear equation generated by
adopting the following step;
(i) Obtained the Taylor series expansion of yn +1 , Ki’s and Hi’s about point (xn,, yn) for i =1(1)R.
(ii) Insert the series expansion into (7).
(iii) Compare the final expansion with Taylor series expansion of yn+1 about (xn, yn) in the power of h.
The number of parameters normally exceeds the number of equations, but these parameters are choosen to
ensure that (one or more of the following conditions are satisfied).
1. Adequate order of accuracy of the scheme (King 1966).
2. Minimum bound of local truncation error (Gill, 1951).
3. The method has maximum interval of Absolute stability (Blum 1952).
4. Minimize computer storage facilities.
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Vol.3, No.1, 2013

2.1 One Stage Scheme
The general one-stage Implicit Rational R-K scheme is of the form
y n + W1K1
y n +1 = (10)
1 + y n V1H1
where,
K1 = hf ( xn + c1h, y n + a11K1 )
H1 = hg ( xn + d1h, z n + b11H1 ) (11)
g (x n z n ) = − Z n2f(x n , yn ) (12)
c1 = a11
d1 = b11 (13)
Adopting binomial expansion theorem on the RHS of equation (10) and ignoring higher order terms, yields
yn +1 = yn + W1K1 − yn V1H1 + (higher order terms )
2
(14)
The Taylor series expansion of yn+1 gives
h2 h3 2 h4 3
y n +1 = yn + hf n +
2!
Df n +
3!
(
D f n + fyDfn +
4!
) (
D f n + f y D 2f n ) - 3Df n Df y + f y2 Df n + 0h5 )
(15)
where,
Dfn = fx + fnfy
D2fn = fxx + 2fnfxy + 2fnfxy + f n2 fyy
2 3
D3fn = fxxx + 3fnfxxy + 3 f n fxyy + f n fyyy
Similarly expand K1 about (xn, yn) we have,
K1 = hA1 + h2B1 + h3D1 + 0h4 (16)
where,
A 1 = fn B1 = C1Dfn
2 2
D1 = C (Dfnfy+ ½ D fn)
1 (17)
In a similar manner, expansion of H1 about (xn, yn) yields
H1 = hN1 + h2M1 + h3R1 + 0h4 (18)
where,

− fn - d1  2f 2 
N1 = , M1 = 2   Df n + n 
2
yn yn  yn  
− d12  − 2 f n  f 2   
R1 = 2

   
2f
+ f y  Df n + n  + 1 2  D 2 f n − n f nn + f x 
2
 ( (19)
yn  yn
  yn    yn 
Adopting (16) and (18) in (14), we obtained
( ) (
y n +1 = y n + W1 hA1 + h 2 B1 + h3 D1 + 0h 4 − y 2 hN1 + h 2 M1 + h3 R1 + 0h 4
n )
= y (W A − y V N )h + (W B - y V M )h + (W D − y V R )h + 0(h 4 )
2 2 2 2 3
n 1 1 n 1 1 1 1 n 1 1 1 1 n 1 1 (20)
2/n
Comparing the coefficient of the powers of h and h equations (15) and (20) and substitute (17) and (19) to get

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Vol.3, No.1, 2013

W1 + V1 = 1 (21)
W1C1 + V1d1 = ½
With the constraints (13), we obtained family of one stage scheme of order two.

(i) W1 = 0, V1 = 1, c1 = d1 = ½ , a11 = b11 = ½
scheme (10) yields
yn
y n +1 = (22)
1 + y n H1
where H1 = hg (xn + ½ h, Zn + ½ H1).
Also with
(ii) V1 = W1 = ½ , c1 = a11 = ¾, d1 = b11 = ¼ .
The scheme (10) result into
y n + 1 K1
y n +1 = 2
(23)
yn
1 + H1
2
where
K1 = hf (xn + ¾ h, yn + ¾ K1)
H1 = hf(xn + ¼ h, zn + ¼ H1)
Also with
(iii) W1 =1, V1 = 0, c1 =d1 = ½ , a11 = b11 = - ½ .
Scheme (10) result into
yn+1 = yn + K1
where,
K1 = hf(xn + ½h, yn + ½ K1)
Which coincide with Implicit Euler’s Scheme of order 2.

2.2 Two Stage Schemes
The general two-stage implicit of Rational Runge-Kutta scheme is of the form
y n + W1K1 + W2 K 21
y n +1 = (25)
1 + yn (V1H1 + V2 H 2 )
where
 2 
Ki = hf  x n + ci h, y n + ∑ aij k j , i = 1(1)2


 j =1 
 2 
Hi = hg x n + d i h, z n + ∑ bij H j , i = 1(1)2
  (26)
 j =1 

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Vol.3, No.1, 2013

Adopting the same procedure as in one-stage scheme, we obtained the following system of equation for family of
two-stage schemes of order three.

W1 + W2+V1+V2 = 1
W1C1 + W2C2+ V1d1+V2d2 = ½
W1 (a11C1 +a12c2) + W2 (a21c1 + a22c2) + V1 (b11d1 + b12d2) +V2(b21d1 + b22d2) = 1/6

W1C1 + W2C2 + V1d1 + V2d 2 =
2 2 2
2
1
3 (27)
a11 + a12 = c1
a21 + a22 = c2
b11 + b12 = d1
b21 + b22 = d2 (28)
Solving these equations (27 & 28) we obtained family of two stage implicit rational R-K schemes of order three.
(1) W1 = W2 = 0, V1 = ¼, V2 = ¾ , c1 = d1 = a12 = b12 = 1
a11 = b11 = a21 = b21 = 0
c2 = d2 = a22 = b22 = 1 (29)
3

yn
yn +1 =
1+
yn
4
(H 1 + 3 H 2 ) )
where,
H1= hg(xn +h, zn +H2)
H2 = hg(xn + 1/3h, zn + 1/3H2)

(2) Also by setting the values of the parameters we obtain
V1 = W1 = 0, V2 = W2 = ½ , c1 =d1 = 0, c2 = ½ + 3
6

v 3
1+ 3
d2 = 1
2 −
, a 22 = b22 = 13 , a 21 =
6 6
b11 = a11 = 3 12 = a12 = 3
1 ,b −1

equation (25) yields
yn + 1 2 K 2
yn +1 = (30)
y
1+ n H2
2
where
K1 =hf(xn, yn + 1 K1 – 1 K2)
3 3

 3 1+ 3 
K2 = hf  x n + ( 1 2 +

h, y n +  ) K1 + 13 K 2 
 6

 6



H1 = hg (xn, zn + 1/3 H1 = 1/3 H2)

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Vol.3, No.1, 2013


H 2 = hg  x n + ( 1 2 +
3
h, Zn +
1+ 3 (
K1 + 1 3 H 2 )
) (31)
 
6 6

Imposing condition Tn+1 = 0(h5)
We obtain the following equations of two stage family of order four.
V1+V2+W1+W2 = 1
W1c1+W2c2+V1d1+V2d2 = ½
W1c12 + W2c2 + V1d12 + V2 d 22 =
2 1
3

W c +W c +V d +V d =
3
1 1
3
2 2
3
1 1
3
2 2
1
4
W1 (a11c1+a12c2) + W2 (a21c1+a22c2) + V1 (b11d1+b12d2) + V2(b21d1 +b22d2) = 1 6
W1c1 (a11c1+a12c2) + W2c2 (a21c1+a22c2)+ V1d1(b11d1+b12d2) +V2d2(b21d1 +b22d2)= 1 4
W1 (a11c12 + a12c2 ) + W2 (a21c12 + a22c2 ) + V1 (b11d12 + b12d 2 ) + V2 (b21d12 + b22d 2 ) = 1 2
2 2 2 2

W1 = [a11 (a11c1 + a12c2 ) + a12 (a21c1 + a22c2 ) + W2 [a21 (a11c1 + a12c2 ) + a21 c1 + a22c2 )] +
(32)
V1 (b11 (b11d1 + b12 d 2 )b12 (b21d1 + b22 d 2 ) + V2 (b21 (b11d1 + b12 d 2 ) + b22 (b21d1 + b22 d 2 )] = 1
24
With the equations (28)and (32). Possible family of two-stage schemes of order four are obtained by setting

(1) V1 = V2 = 0, W1 = W2 = ½ , d1=c1 = ½ + 3
6

3
d 2 = c2 = 1
2 −
, a 22 = b22 = b11 = a11 = 1
4
6
3 3
a a12 = b12 = 1 4 + , a 21 = b 21 = 1 4 −
6 6
These into equation (25) yields
yn+1 = yn + ½ (K1 + K2)

 3  3
where K1 = hf (xn +  12 + h, y n + 1 4 K1 +  1 4 + K2
 6   6 
   
 3 3
 2 + 6 h + y n + ( 4 − K1 − 6  K1 + 4 K 2 )
K2 = hf (xn + 1  1  1 (33)
  
Which incidentally coincide with 2-stage Implicit R-K scheme of order four. Proposed by Harmmer and Holling
Worth (1955).

(ii) W1 = W2 = 0, V1 = V2 = ½, c2 = d2 = ½ - 3
6

a11 = b11 = a22 = b22 = ¼ , a12 = b12 = ¼
Equation (25) yields
yn
yn +1 = (34)
y
1 + n ( H1 + H 2 )
2
 3  3
H1= hg (xn +  12 + , z n + 1 4 H1 +  1 4 + H 2 )
 6   6 
   

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Vol.3, No.1, 2013

 3 3
 2 − 6 , z n + ( 4 − 6  H1 + 4 H 2 )
H3 = hg (xn + 1  1  1

  
Next section analyses the error, consistency, convergence and stability property of these schemes.

3. Error, Convergence and Stability Properties
In this section, we shall consider the error, convergence, consistency and stability properties of these schemes.

3.1 Error Analysis
Error of numerical approximation techniques for Stiff ODEs arise from different causes that can be majorly
classified into discretization, truncation, and round –off error respectively.
Round-off error is an error introduced as a results of the computing device. Mathematically it can be expressed as
Yn +1 = yn +1 − Pn +1 (35)
where yn+1 is the expected solution of the difference equation (10), while Pn+1 is the computer output at (n+1)th
iteration.
Truncation error on the other hand is the error introduced as a result of ignoring some of the higher terms of the
power series (Taylor and Binomial series expansion) during the development of the new schemes.
Discretization error en+1 associated with the formular (10) is the difference between the exact solution y(xn+1) and the
numerical solution yn+1 generated by (10) at point xn+1. That is
en+1 = yn+1 – y(xn+1) (36)

3.2 Consistency Property
The one-step scheme is said to be consistent if
y n +1 − yn
lim = f(x n , y n ) (37)
h →o h
To show the consistency, we recall that
R R
y n +1 = yn − y 2 ∑ Vi H i + ∑ Wi K i + (Higher order terms)
n (38)
i =1 i =1
Subtract yn from both sides and ignoring higher order terms
R R
y n +1 − yn = ∑ Wi K i − yn ∑ Vi H i
2
(39)
i =1 i =1
Substituting the expression for Hi and Ki in equation (8)
R
 j
  i 
y n +1 − yn = ∑ Wi hf 
 xn + c1h1 yn + ∑ a ij K j  + − yn ∑Vi hg  xn + d i h, zn + ∑ bij H j  (40)

2
 
i =1  i =1   j =1 
Dividing by h and taking limit as h → o
yn +1 − yn R
lim = ∑ W1 f (x n, , yn ) − yn ∑Vi g ( xn , zn )
2
(41)
h→o h i =1

y − yn R
lim n +1 = ∑ (W1 + V1 ) f ( xn , yn ) (42)
h →o h i =1

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Vol.3, No.1, 2013

yn +1 − yn
∴ lim = f ( xn , yn ) (43)
h →o h
This shows that Implicit Rational R-K scheme is consistent.
According to Lambert (1963), a consistent one-step method is convergent. Hence the new scheme is convergent.

3.3. Stability Property
To show the stability of the scheme, we apply (10) to Dahlquist (1963) stability scalar test initial value problem.
y′ = λy, y(xo ) = yo (44)
For example, the stability scheme (34) with
V1 = V2 = ½, W1 = W2 = 0, c1 = d1 = ½ - 3 ,
6
c2 = d2 = ½ + 3 , b11 = b22 = a11 = a22 = ¼ .
6
b12 = a12 = ¼ + 3 , a21 = b21 = ¼ - 3
6 6
is
1 + 12 Z + 12 Z 2
µ (Z ) = (45)
1 − 12 Z − 512 Z 2
This scheme is A-stable with ( -∞, 0) as interval of Absolute stability. Since
lim it µ ( Z ) < 1 (46)
Z →∞

3.4 Numerical Computations and Results
In order to access the performance of the schemes the following sample problem were solved.

Problem 1:
Consider the Stiff systems of ODEs
Y′ =AY (47) Where A
1.0 − 4.99 0 
=
0 − 5.0 0  (48)
 
0
 2.0 − 12

with initial condition y(o) = (2, 1, 2), 0≤ x ≤ 1
Using step size h = 0.01, the method is implemented and the results are shown in Table (1).

Problem 2:
The second sample problem considered is the Stiff system of initial values problem in ODEs.

 - 0.5 0 0 0   y1 
 
 0 - 1.0 0 0  y2 
 
y′ =  (49)
0 0 - 9.0 0  y3 
  
 0 0 0 - 10.0   y 4 
 
With initial condition y(o) = [1 1 1 1] ,
The results are shown in Table 2.
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Vol.3, No.1, 2013

4. Conclusion
Implicit Rational Runge-Kutta method for the integration of Stiff system of ODEs has been proposed. Theoretically
it has been showed that the scheme is consistent, convergent and A – stable. Numerical results showed that the
scheme is accurate and effective. Also from the above results the error is very minimal and this implies that the
scheme is very accurate.

References
Babatola, P.O. (1999), “Implicit Rational R-K Scheme for Stiff ODEs” M.Tech. Thesis, Federal University of
Technology, Akure.
Blum, E.K. (1952), “A modification of Runge- Kutta Fourth-order Method”, Maths Comp, Vol. 16, Pg. 176 – 187.
Curtis, C.F. and J.O. Hirschfelder (1952), “Integration of Stiff Equations. “National Academy of Sciences, Vol. 38
pg. 235 – 243.
Dalhquist, D. (1963), “A special stability problem for Linear Multi-Step methods” BIT 3, Pg. 27 – 43.
Fatunla, S.O. (1982), “Numerical Integrator for Stiff and Highly Scillatory problem in Differential Equation Maths
Computations Vol. 34 Pg. 374 – 400.
Fatunla, S.O. (1987), “Recent Advances in ODEs. Proceeding of the First International Conference on Numerical
Analysis, Pg. 27 – 31.
Gill, S. (1951), “A process of step by step Integration of Differential Equation in an Automatic Digital Computing
Machine”. Proc. Cambridge Philos Soc. Vol. 47, pg. 95 – 108.
Hammer, P.C. and Hollingsworth, J.W (1955), “Trapezoidal methods of Approximating solution of ODEs”,
M.T.A.L. Vol. 9 pg. 92 – 96.
Hong Yuanfu (1982), “A class of A-stable or A(α) stable Explicit Schemes” Computational and Asymptotic Method
for Boundary and Interior Layer” Proceeding of BAILJI Conference Trinity College, Dublin Pg. 236 – 241.
Jain, M. K. (1979), Numerical Solution of ODEs, Wiley Eastern Limited, Pg. 200 – 210.
King, R. (1966), “Runge-Kutta methods with constraints Minimum Error Bound”, Math Comp Vol. 20, Pg. 386 –
391.
Lambert, J.D (1963), “On the numerical solution of y′ = f ( x, y) by A Class of Formular Based on Rational
Approximate”, Maths Comp.Vol 19, Pg. 456 – 462.
Okunbor, D.F (1985), “Explicit RR-K schemes for system of ODEs” MSc Thesis, University of Benin, Benin City
(Unpublished).

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Vol.3, No.1, 2013

TABLE 1: NUMERICAL RESULT OF A - STABLE IMPLICIT RATIONAL RUNGE-KUTTA SCHEMES FOR
SOLVING STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
Y1 Y2 Y3
CONTROL STEP
Xn E1 E2 E3
SIZE h
.1980099667D+01 .9706425830D+00 .8869204674D+00
.3000000000D – 01 .3000000000D – 01 .8291942688D-09 .3281419103D-07 .8161313500D-05
.1885147337D+01 .8379203859D+00 .4917945068D+00
.1774236000D+00 .1771470000D-01 .9577894033D-01 .3422855333D-08 .5357828618D-06
.1791235536D+01 .7191953586D+00 .2663621637D+00
.3307246652D+00 .1046033532D-01 .11050933794D-10 .35587255336D-09 .3474808041D-07
.1694213422D+01 6088845946D+00 .1365392880D+00
.4977858155D+00 .6176733963D-02 .1269873096D-11 .3655098446D-10 .2146555961D-08
.1556933815D+01 .4729421983D+00 .4953161076D-01
.7512863895D+00 .3647299638D-01 .1425978891D-08 .3505060447D-07 .1010194837D-05
.1435390902D+01 .3709037123D+00 .1867601194D-01
.9951298893D+00 .2153693963D-01 .1594313570D-09 .3316564301D-08 .4481540687D-07

TABLE 2: NUMERICAL RESULT OF A-STABLE IMPLICIT RATIONAL RUNGE-KUTTA SCHEMES FOR
SOLVING STIFF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
Y1 Y2 Y3 Y4
CONTROL STEP
Xn E1 E2 E3 E4
SIZE h
.9950124792D+00 .9900498337D+00 .9139311928+00 .9048374306D+00
.3000000000D – 01 .3000000000D – 01 .2597677629D-10 .4145971344D-09 .2617874150D-05 .3971726602D-05
.9708623323D+00 .9425736684D+00 .5872698932D+00 .5535451450D+00
.1774236000D+00 .1771470000D-01 .3078315380D-11 .4788947017D-10 .2005591107D-06 .2890213078D-06
.9402798026D+00 .8841261072D+00 .3300866691D+00 .2918382654D+00
.3694667141D+00 .1046033532D-01 .3621547506D-12 .5454525720D-11 .1355160001D-07 .1829417523D-07
.9144602205D+00 .8362374949D+00 .1999708940D+00 .1672231757D+00
.5365278644D+00 .6176733963D-02 .4285460875D+13 .6268319197D-12 .9915873955D-09 .1265158728D-08
.8693495443D+00 .7557686301D+00 .8044517344D-01 .6079796167D-01
.8400599835D+00 .3647299638D-01 .4961209221D-10 .6922001861D-09 .5087490103D-06 .5899525189D-06

16

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A new class of a stable implicit schemes for treatment of stiff

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A new class of a stable implicit schemes for treatment of stiff