Implicit Runge-Kutta Method for Van Der Pol Problem

(1)

Published online July 13, 2014 (http://www.sciencepublishinggroup.com/j/acm) doi: 10.11648/j.acm.s.2015040101.12

ISSN: 2328-5605 (Print); ISSN: 2328-5613 (Online)

Implicit Runge-Kutta method for Van der pol problem

Jafar Biazar

*

, Meysam Navidyan

Department of Applied Mathematics, Faculty of Mathematical sciences, University of Guilan, P.O Box: 41635-19141, Rasht, Iran

Emails address:

[email protected] (J. Biazar), [email protected] (J. Biazar), [email protected] (M. Navidyan)

To cite this article:

Jafar Biazar, Meysam Navidyan. Implicit Runge-Kutta Method for Van Der Pol Problem. Applied and Computational Mathematics.

Special Issue: New orientations in Applied and Computational Mathematics. Vol. 4, No. 1-1, 2015, pp. 6-11. doi: 10.11648/j.acm.s.2015040101.12

Abstract:

In this manuscript the implicit Runge-Kutta (IRK) method, with three slopes of order five has been explained, and is applied to Van der pol stiff differential equation. Truncation error, of order five, has been estimated. Stability of the procedure for the Van der pol equation, is analyzed by the Lyapunov method. To illustrate the structure of the method, an Algorithm is presented to solve this stiff problem. Results confirm the validity and the ability of this approach.

Keywords:

Implicit Method, Taylor Series, Legendre Orthogonal Polynomial, Van Der Pol Equation, Lyapunov Function

1. Introduction

An Implicit Runge-Kutta method for solving differential equation y′ =f t y( , ) with ν slopes is defined by the following equation:

1

n n i i

i

y

w K

ν

+

=

+

∑

Where

1

( , )

i n i n i j j

j

K h f t c h y a K

ν =

= + +

∑

and

1

, 1, 2, ..., i ij

i

c a i

ν

=

∑

=

And a_{i j} , 1 ≤ i , j ≤ ν , w ₁,w ₂, . . . ,w _ν are parameters that will be determined. The function

K

_j is defined by a set of

ν

implicit equation.( see [1] and [6] )

In electronic, the Van der Pol oscillator is a

non-conservative oscillator with non-linear damping. This problem was originally introduced by Van der pol (1926) in the study of electronic circuit by the following second order stiff differential equation

2

''

(

1) '

0. u

+

ε

u

−

u

+ =

u

Where

u

is a function of the time

t

,

and

ε

is a positive scalar parameter indicating the nonlinearity and the strength of the damping. (see [4] p-121 & [1] p-16)

2. The Implicit Runge-Kutta Method

In this implicit method let's

ν

=

3

, then we have the following equation

1 1 2 2 3 3

1 1 1 2 2 3 3

1 2 3

( , ),

, , 1, 2,3 .

i n i n i i i

n n

i i i i

K h f t c h y a K a K a K

y y w K w K w K

c a a a i

+

= + + + +

 

= + + +



 ₌ ₊ ₊ ₌



(1)

Where

a

_i₁

,

a

_i₂

,

a

_i₃

,

w

_i

,

i

=

1, 2, 3

are twelve arbitrary parameters, which should be determined. The Taylor series gives

2 3 4

1

1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ...

2 6 2 4

iv

n n n n n n

(2)

2 2

2 3 2

( ) ( , )

( ) ( ) ( )

( ) [( 2 ) ( )]

( ) [( 3 3 ) ( 2 )

( )(3 3 )] 3

n n n

n t y n n

n tt ty y y y t y y

iv

n ttt tty ty y y y y y tt ty y y

t y ty y y y n y y y

y t f t y

y t f f f f f D f

t y

y t f f f f f f f f f D f f D f

y t f f f f f f f f f f f f f

f f f f f f f D f f D f D f D f f

′ =

∂ ∂

′′ = + = + =

∂ ∂

′′′ = + + + + = +

= + + + + + +

+ + + + = + + + 2

D f

Based on expansion of two-variable function, the equation

K

_i in (1) changes to:

(3)

These equations are implicit and we cannot easily obtain the explicit expression forK₁, K₂, andK₃. In order to determineK K₁, ₂,andK₃explicitly, we assume the following form

(4) Where

A

_i

,

B

_i

,

E

_i

,

D

_i, and

F

_i are unknown to be determined.

Substituting forK K₁, ₂,and K₃from ( 4 ) into (3) , and on equating the terms with identical powers of

h

in Taylors series, we obtain the following results:

1 1 2 2 3 3

2 2 1 1 2 2 3 3

1 11 1 12 2 13 3 2 2 1 1 22 2 2 3 3 2

3 3 1 1 3 2 2 33 3 1 1 2 2 3 3

1

( ) ( )

1

( )

2

[ ( ) ( )

( )] ( )

1 ( 2 i n

i i t i i i y i t n y i

i i i i y i

i i i

i y n i i i i n y

i

A f

B c f a A a A a A f c f f f c D f

E a c a c a c f D f c D f

D a a c a c a c a a c a c a c

a a c a c a c f D f c a c a c a c D f D f

a

=

= + + + = + =

= + + +

= + + + + +

+ + + + + +

+ 2 2 2 2 3 3

1 2 2 3 3

1 )

6

i i y n i n

c +a c +a c f D f + c D f

1 11 11 1 12 2 13 3 12 21 1 22 2 23 3

2

13 31 1 32 2 33 3 1 11 1 12 2 13 3

2 2 2 2 3 3

11 1 12 2 13 3 1

2 21 11 1 12 2 13 3 22 21 1 22 2

[

[(

(

)

(

)

(

))

(

)

1

1 (

)

]

2

6 [(

(

)

(

i i

y n n y

y n n

i

F

a

a c

a

a c

a

a c

f

Df

c a c

a c

a c Df Df

a c

f D f

c D f

a

a c

a

a c

a

=

+

23 3

2

23 31 1 32 2 33 3 2 21 1 22 2 23 3

)

(

))

_y _n

(

)

_n _y

c

a

a c

f

Df

c a c

a c

a c Df Df

(3)

2 2 2 2 3 3

21 1 22 2 23 3 2 3 31 11 1 12 2 13 3

2 32 21 1 22 2 23 3 33 31 1 32 2 33 3

2 2 2 2 3 3

3 31 1 32 2 33 3 31 1 32 2 33 3 3

1 1 ( ) ] [( ( ) 2 6 ( ) ( )) 1 1 ( ) ( ) ]] 2 6 [

y n n i

y n

n y y n n y

i

a c a c a c f D f c D f a a a c a c a c

a a c a c a c a a c a c a c f Df

c a c a c a c Df Df a c a c a c f D f c D f f

c

+ + + + + + +

+ + + + + +

+ + + + + + +

+ 2

1 11 1 12 2 13 3 1

2 2 21 1 22 2 23 3 2

2 2 2

3 31 1 32 2 33 3 3 1 1 2 2 3 3

2 2

1 1 2 2 3 3 1 1 2 2 1 (( ) ) 2 1 (( ) ) 2 1 1 (( ) )] ( ) 2 2 1 ( ) ( 2

i y n n

i y n n ty i i i yy n

i i i i tty n i i i

a a c a c a c f Df c Df

a a c a c a c f Df c Df f a c a c a c f D f

c a c a c a c f Df c a c a c a

+ + +

+ + + +

+ + + + + + +

+ + + + + + 3 3

2 2 4 4

1 1 2 2 3 3

)

1 1

( ) , 1, 2, 3

2 24

i n tyy n

i n i i i yyy n i n

c f f Df

c f a c a c a c f Df c D f i

+ + + + =

(5)

Numerical method in equation

(1)

with the help of

(4)

may be written as

2

1 1 1 2 2 3 3 1 1 2 2 3 3

3 4

1 1 2 2 3 3 1 1 2 2 3 3

( ) ( )

( ) ( ) ...

n n

y y h w A w A w A h w B w B w B

h w E w E w E h w D w D w D

+ = + + + + + +

+ + + + + + + (6)

Where A B E Di, i, i, i, and Fi are given by.(5)

By equating the coefficients of the terms with the identical powers of

h

in

(6)

and

(2)

, the following equations are obtained. which are the same as the system of equations Butcher introduced in ( [1] & [6])

3 1 3 1 3 2 1 3 3 1

(7 ) : 1,

1

(7 ) : ,

2

1

(7 c) : ,

3

1

(7 ) : ,

4 i i i i i i i i i i i a w

b w c

w c

d w c

= = = = = = = =

∑

3 , 1 3 , 1 1

(7 ) : ,

6

1

(7 ) : ,

8

i ij j i j

i i ij j i j

e w a c

f w c a c

= = = =

∑

3 , 1 3 2 , 1 3 , , 1

1

( 7 ) : ,

8 1

( 7 h ) : ,

1 2 1

( 7 i ) : .

2 4

i i i j j i j

i i j j i j

i i j j k k i j k

g w c a c

w a c

w a a c

= = = = = =

∑

i

w

,

i

=

1, 2, 3

can be determined from (7a−7 )d , and

from (7e−7 )i and three more the following equations

1 2 3, 1, 2,3

i i i i

c =a +a +a i= , ai j, ,i j=1, 2, 3 will be calculated.

Butcher (1964) introduced RK method based on the Radau and Lobatto quadrature formulas. In this procedure the coefficients

c

_i are taken the Radau's roots of the Legendre polynomial of degree three:

( )

(

)

2

3 2

1 2 3

2

4 6 4 6

1 0 , , 1 .

1 0 1 0

d

x x c c c

d x

− +

− = ⇒ ₌ ₌ ₌

The solution of the system

(7)

, after substitution of the values of

c

_i results in the following coefficient of Radau formula of order five:

1 6 6 _{8 8} ₇ ₆ _{2 9 6} _{1 6 9} ₆ ₂ ₃ ₆

4 6

3 6 _{3 6 0} _{1 8 0 0} _{2 2 5}

1 0

1 6 6 _{2 9 6} _{1 6 9} ₆ _{8 8} ₇ ₆ ₂ ₃ ₆

4 6

, _{3 6}

1 8 0 0 3 6 0 2 2 5

1 0

1 6 6 1 6 6 1

1

1 3 6 3 6 9

9

C W A

 ₋  _ _  ₋  _ _ ₋ ₋ _{− +}     _ _     _ _ +    ₊  _ _ ₊ ₊ _{− −} =   = _ _ =       _ _     _ _ ₋ ₊     _ _     __ __  

(4)

(8)

3. Numerical Example

To illustrate the method, let's apply IRK on the following stiff problem, which is known as Van der pol equation:

2

y''+

ε

(y −1) y'+y =0, y(0)=2,y '(0)= 0.

In the first step by considering the new dependent variables, Van der pol equation is written, equivalently, as the following system of two first order differential equation

:

3

' ( , ) ( ), u(0) 2 3

2

' ( , ) , (0)

3

u u f u v v u

v g u v u v

ε



= = − − =

 

 ₌ _{= −} ₌



(9)

For applying iterative formula

( )

8 , to the system

( )

9 the

parameters K_ui,K_vi should be computed from the IRK_NEWTON ALG( ɛ, step size h,

u

₀,

v

0)

following formulas:

, ,

( , , )

ui n i n i j u j n i j v j

j j

vi n i n i j u j n i j v j

j j

K hf t c h u a K v a K

K hg t c h u a K v a K

 ₌ ₊ ₊ ₊

 

= + + +

 

∑

Where

i j

,

=

1, 2,3

. Then

1 1 2 3

1 6 6 1 6 6 1

3 6 3 6 9

1 6 6 1 6 6 1

3 6 3 6 9

n n u u u

n n v v v

u u K K K

v v K K K

+ +

 ₋ ₊

= + + +

  

− +

 ₌ ₊ ₊ ₊



(10)

Pseudo code Algorithm: Let's explain the above method with following Pseudo code of Newton iterative procedure for solving example

( )

9 :( takei j, =1, 2,3)

(5)

equation, are plotted for

ε

=10,h=0.1, in the following Fig1.

Fig. 1. The solution of Vsn der pol equation

4. Truncation Error Analysis

Definition:[6] truncation error is the quantity

T

which

must be added to the computed quantity in order that the result be exactly equal to the quantity that we are looking for. This means:

(

true computed quantit

)

(

)

y y + =T y exact solution

Then the exact value of

y t

( )

will satisfy

1

( n ) ( n) ( n, ( n), ) n

y t ₊ = y t +h

ϕ

t y t h +T

Where

ϕ

( , ( ), )t_n y t_n h is a function of the argument

,

t y

, and

h

, and is called the increment function, and n

T

is the local truncation error. Let's taken 6 ( )

y t = t as

computed quantity of( 8 )we get:

5

1 1 2 3

6 6 6 5 5

1

5 6

( , ) 6

16 6 1 6 6 1

( ) ( )

36 3 6 9

16 6 4 6 1 6 6 4 6

( ) 6 ( ) 6 ( )

36 1 0 36 1 0

1 1

6 ( )

9 1 0 0

n n n

n n n n n

n n n

f t y t

y t y t K K K T

t t h t h t h h t h

h t h T T h

+

=

− +

= + + + +

− − + +

= + = + + + +

+ + + ⇒ _{= −}

The traditional value of the truncation error is usually considered as: 6 ( 6 )

6 ( ) .

n

T =C h y ζ comparing with the above value of truncation error results in

6

1 7 2 0 0 0

C = − .

So the truncation error is of

O h

(

6

)

, i.e. the method

(8)

is of order five. (see [6] and [1])

5. Stability Analysis of the Van Der Pol

System

Definition 1. Stability and asymptotic stability: The solution of a system of equation say X′ =F X

( )

, is stable, if for all

t

≥

t

₀we get: [9]

0 0

0 _ε 0 ( x(t ) x(t ) x t( ) x t( ) )

ε δ δ ε

∀ > ∃ > − < ⇒ − <

and the solution is asymptotic stable, if it is stable and also

0 0 ( x(t )0 x(t )0 0 x(t) x t( ) 0)

δ δ

∃ > − < ⇒ ₋ _→ _{, as}

t

→∞

Definition 2. Invariant set: A set such as n

M ⊂R is

invariant set of system of equation X ′ =F X

( )

, if from any

0

X ∈M conclude that X t X( , 0)∈M, for all

t

∈

R

[9].

Theorem l.If the scalar positive definite function ( ),V X

called Lyapunov function, defined on the set

{

n:

}

p

S = X ∈R X <P , and V X′( )≤0 then the zero solution of X ′ =F X( )is stable [9].

Theorem 2. Let's the positive definite function

( )

V X exists as such V X′( )≤0 on the open set Ω , inX′ =F X

( )

,

F

is a function : n

F Ω →R ,M includes all invariant subsets of E ⊂c_λ , where

{

n : 0

}

E = X ∈R V′= and c_λ =

{

X ∈Rn :V X( )≤

λ

}

.

Then any solution X t X( , ₀)∈C_λ of X′ =F X

( )

, converges into theM [11].

Corollary. Let's the assumptions of the theorem2 hold. If zero is only invariant point of

E

, then the zero solution of X′ =F X( ) is asymptotically stable [11].

Stability of the system of equation

(9)

is proved in the following:

Let's _{( , )} 1₍ 2 2₎

2

V u v = u +v , which is positive definite on

2

R

and

2 2

( , ) ( 1)

3 u

V u v′ = −

ε

u − , then on the strip

{

_{( , )}

_{u v}

_R

2

_:

₃

_u

_3,

_v

}

Ω=

∈

− < <

−∞< <∞

, we have ( ) 0

(6)

'

0 u

v

=





=



Regarding the last corollary the set including zero is the only invariant subset of E.And

( )

0, 0 is the asymptotic

stable point of (9). To determine the asymptotic stability

area, Let's consider the set of curves V u v( , )=λ,where 0, ( , )u v .

λ≥ ∈Ω This set is obviously closed and the curves are symmetric with respect to the

u axis

−

.

The function

2

u _{is decreasing on the interval}

(− 3 , 0 ] and

increases on the 0 < <u 3 . For the constant

λ

,

the curve V u v( , )=

λ

cuts the borders at one of the points (− 3, 0)or( 3, 0). So the best value for the parameter

λ

is

equal to 2 2

( 3 ) ( 3 ) 3

( , )

2 2 2

m in

λ = − = , and asymptotic

stability area consists of the pointes in the closed circle

{

_{( , )} _: 2 2 ₃

}

C_λ= u v ∈Ω u + ≤v . Then any limit cycle of the Van der pol equation is out of the circle u2+v2=3(see [1] p-16).

6. Conclusions

Since there are three different topics studied, conclusion is also divided in different parts;

A-From truncation error section, we conclude that order of this implicit method is five, and this means that this numerical method is precise for polynomials of degree less than six.

B- The disadvantage of the Runge-Kutta methods is that they involve considerably more computations, but have the advantage of self starting.

C- Method

( )

_{1 0} with

h

=

0.1

can only used until

33

ε = , and for ε>33, the order of the method should increased or related step size decreased.

D- From stability analysis section, we conclude that the

method applied to the Van der pol equation is stable, And this means the formula of the numerical method is insensitive to small change in the local errors.

References

[1] Butcher J.C. Numerical Methods for Ordinary Differential Equations. John Wiley,2003.

[2] Frank R, Schneid J, Uberhuber C.W: Order results for implicit Runge-Kutta methods applied to stiff systems. SIAM J. Numer. Anal., 22, 515-534 (1985).

[3] Hairer E, Lubich C, Roche M: Error of Runge-Kutta methods for stiff problems studied via differential algebraic equations. BIT 28, 678-700 (1988).

[4] Hairer E, Lubich C, Roche M: The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Springer verlag (1989).

[5] Hairer. E, Wanner. G & Nørsett S.P. " Solving Ordinary Differential Equations I, nonstiff problems ", Springer Series in Computational Mathematics 14, DOI 10.1007/978-3-642-05221-73, © Springer-Verlag Berlin Heidelberg 2010. [6] Jain M.J. Numerical Solution of Differential Equations.

John Wiley & Sons (Asia) Pte Ltd(1979).

[7] Kalman R. E. & Bertram J. F: "Control System Analysis and Design via the Second Method of Lyapunov", J. Basic Engrgvol.88 1960 pp.371; 394.

[8] Lefschetz.s. Differential equation: Geometric theory, 2nd edition. Interscience, New York, 1963.

[9] Lakshmikantham v, Leela s: Differential and integral inequalities: theory and applications, volume I, Acalemic Press.(1969)

[10] Rama Mahana Rao.M. A note on an integral inequality, J. Indian Math, Soc. 27, 67-69, 1963.