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The extrapolation technique has been proved to be very powerful in improving the performance of the approximate methods by large time whether engineering or scientific problems that are handled on computers. In this paper, we investigate the efficiency of extrapolation of explicit general linear methods with Inherent Runge-Kutta stability in solving the non-stiff problems. The numerical experiments are shown for Van der Pol and Brusselator test problems to determine the efficiency of the explicit general linear methods with extrapolation technique. The numerical results showed that method with extrapolation is efficient than method without extrapolation.

The two traditional numerical methods such as linear multi-step and RK methods had been studied separately to solve the problems in ordinary differential equations. General linear methods (GLMs) have been considered by Butcher as a unifying framework for both multi-value and multi-stage methods [

General linear methods are constructed by four coefficient matrices A, U, B and V utilized in partitioned ( s + r ) × ( s + r ) matrix, given by

[ A s × s U s × r B r × s V r × r ]

The coefficients of these matrices denote the connection between various numerical quantities that arise in the computation. The coefficient matrix A denotes the implementation costs of GLM methods.

Consider the initial-value problem given by

y ′ ( x ) = f ( y ( x ) ) , x ∈ [ x 0 , X ] y ( x 0 ) = y 0 (1)

The general linear methods for N-dimensional differential Equations (1) are given as follows:

Y i = ∑ j = 1 s a i j h f ( Y j ) + ∑ j = 1 r u i j y j [ n − 1 ] , i = 1 , 2 , ⋯ , s , y i [ n ] = ∑ j = 1 s b i j h f ( Y j ) + ∑ j = 1 r v i j y j [ n − 1 ] , i = 1 , 2 , ⋯ , r , (2)

where Y i known as internal stage value, f ( Y j ) known as the identical stage derivatives, y i [ n ] and y i [ n − 1 ] are known as the incoming approximation and outgoing approximation respectively.

For more convenience, the above formula can be written in compact form as follows:

Y [ n ] = h ( A ⊗ I m ) F ( Y [ n ] ) + ( U ⊗ I m ) y [ n − 1 ] ,

y [ n ] = h ( B ⊗ I m ) F ( Y [ n ] ) + ( V ⊗ I m ) y [ n − 1 ] ,

where

Y [ n ] = [ Y 1 [ n ] Y 2 [ n ] ⋮ Y s [ n ] ] , F ( Y [ n ] ) = [ F ( Y 1 [ n ] ) F ( Y 2 [ n ] ) ⋮ F ( Y s [ n ] ) ] , y [ n − 1 ] = [ y 1 [ n − 1 ] y 2 [ n − 1 ] ⋮ y s [ n − 1 ] ] .

The components of input vector y i [ n − 1 ] of next step satisfied as follows

y i [ n − 1 ] = ∑ k = 0 p q i , k h k y ( k ) ( t n − 1 ) + O ( h p + 1 ) , i = 1 , ⋯ , r ,

given for some parameters q i , k where i = 1 , 2 , ⋯ , r and k = 0 , 1 , ⋯ , p .

The GLMs have order conditions p if the components of output vector y i [ n ] satisfied as follows

y i [ n ] = ∑ k = 0 p q i , k h k y ( k ) ( t n ) + O ( h p + 1 ) , i = 1 , ⋯ , r ,

given also for same the parameters q i , k .

The quantities y ( n ) , y ( n − 1 ) , Y and f ( Y ) of GLMs can be related as

[ Y y ( n ) ] = [ A ⊗ I U ⊗ I B ⊗ I V ⊗ I ] [ h f ( Y ) y ( n − 1 ) ]

There are some known methods that can be formulated as general linear methods given in the following [

First consider the RK methods given by

0 1 2 1 2 1 2 0 1 2 1 0 0 1 1 6 1 3 1 3 1 6

are written as formula of GLMs as follows:

[ 0 0 0 0 1 1 2 0 0 0 1 0 1 2 0 0 1 0 0 1 0 1 1 6 1 3 1 3 1 6 1 ]

Another example of Adams-Bashforth method that is given by

y n = y n − 1 + h ( 3 2 f ( y n − 1 ) − 1 2 f ( y n − 2 ) ) ,

can be written as formula of GLMs as follows:

[ 0 1 3 2 − 1 2 0 1 3 2 − 1 2 1 0 0 0 0 0 1 0 ]

The researchers have been trying many ways to construct the efficient GLMs in solving the ordinary differential equations (ODE), such as [

In this paper, the advantages of using the extrapolation technique with explicit GLMs methods are given for some non-stiff problems. The extrapolation technique provides one of the essential sorts of the numerical integrator to solve the ordinary differential equations within an efficient step-size control mechanism and uncomplicated variable order strategy.

The organization of this paper is as follows. Section 2 discusses the order conditions of GLMs with IRKs property, where the order conditions p considered is equal to stage order q. The construction of GLMs with IRKs property is given in Section 3. Section 4 explains the types of extrapolation technique such as active and passive. Although there are two modes in applying extrapolation, this article only considers passive extrapolation to solve some non-stiff problems such as Van der Pol and Brusselator test problems. The numerical results are given in Section 5.

There are challenges between the researchers to construct and implement the GLMs in an easy way. Most of them consider two main assumptions in constructing these methods.

First assumption is by denoting the order condition p equal to the stage order q. The stage values satisfied as follows:

Y [ n ] ≈ [ y ( x n − 1 + c 1 h ) y ( x n − 1 + c 2 h ) ⋮ y ( x n − 1 + c s h ) ] + O ( h p + 1 ) . (3)

Second assumption denotes the quantities should have a simple structure passed from step to step in order to avoid the complicated of changing step-size. Therefore, the Nordsiek form is required to present the input and output quantities.

y [ n ] ≈ [ y ( x n ) h y ′ ( x n ) ⋮ h p y ( p ) ( x n ) ] + O ( h p + 1 ) (4)

In order to describe the order condition, then the following theorem is needed [

Theorem 1 GLMs with coefficient matrices A, U, B and V represented in Nordsieck representation, has p = q if and only if

exp ( c z ) = z A exp ( c z ) + U Z + O ( z p + 1 ) ,

exp ( z ) Z = z B exp ( c z ) + V Z + O ( z p + 1 ) ,

where exp ( c z ) indicates the vector for ith components which is equal to exp ( c i z ) . This means

exp ( c z ) = [ exp ( c 1 z ) exp ( c 2 z ) ⋮ exp ( c s z ) ] .

Construction GLMs with Inherent RK stability that was given by Will Wright [

In order to make the stability of GLMs similar with RK methods, first consider all the eigenvalue of the stability matrix M ( z ) equal to one, which is given by

M ( z ) = V + z B ( I − z A ) − 1 U ,

and consider the stability function

The stability function can be solved by assuming

The following definition is important to show the idea of IRKs for GLMs.

Definition 1 [

The general linear methods have stability as same as RK methods if its stability function

where

Moreover, there is another important definition necessary for the IRKS property.

Definition 2

The general linear methods, which satisfy

where J denotes the shifting matrix, which is given by

The following theorem discusses the GLMs with IRKs in a more practical way [

Theorem 2

If the general linear methods have IRKs property with stability matrix considered as

Then the stability polynomial

Proof.

Consider another stability matrix that have the same property of the old one that is given by

By applying to the system of Equations (5) into the right side of (6)

Then, the matrix V is similar to

Now, by assume the matrix V has only one non-zero eigenvalue to guarantee the methods is stable, then the stability matrix

In [

• Choose the constant

so that the initial value problem with stability function

• Choose the vector c as

• Choose the parameters

• Define the following coefficients

• Find LU decomposition of the coefficient

• Consider the matrix

Therefore, they are considered the basic coefficients of explicit GLMs with inherent RK methods as follows:

Richardson [^{2}-asymptotic error expansions and generalized the concept of symmetry to composite RK methods that preserve the h^{2}-error expansion and also create the necessary damping. These methods have same upside to increase the order by two at a time on successive extrapolations. Later, Gorgey in [

In [

Therefore, the extrapolation technique has an efficient way to improve the accuracy of GLMs with inherent RK stability. We use MATLAB code irks14-extrap.m that has extrapolation implementation of explicit GLMs with inherent RK stability.

In this section, the results of numerical experiments obtained by the MATLAB code irks14-extrap.m are discussed. This code based on GLMs with inherent RK stability of order

These results will show which one the most efficient methods to solve the non-stiff problems such as brusselator and Van der Pol test problems.

The first test problem is Brusselator (BRUS) which is an autocatalytic oscillating chemical reaction equation. It is a system of two ordinary differential equations which is assumed as follows

where

The second test problem is Van der Pol (VDP) which is defined as follows:

where the first equation denotes the non-stiff equation while the second equation denotes stiff equation with including small

In this paper, we discuss some issues related to developing the construction of Explicit general linear methods with inherent Runge-stability for numerical solution of non-stiff differential equations. These issues include the application of extrapolation technique to this base method. As we can see from the results in this paper, the extrapolation technique with explicit GLMs with inherent RK stability of order four on BRUS and VDP test problems, both give better efficiency than explicit GLMs without extrapolation.

Future work will involve verifying the efficiency of extrapolation of explicit GLMs with inherent RK stability for higher order for numerical solution of non stiff differential equations as well as the implicit GLMs for stiff equations.

The authors would like to extend their gratitude to the Sultan Idris Education University especially the Research Management and Innovation Center for providing the research grant GPU, Vote No: 2018-0149-102-01.

The authors declare no conflicts of interest regarding the publication of this paper.

Kadhim, A.J. and Gorgey, A. (2019) Extrapolation of GLMs with IRKS Property to Solve the Ordinary Differential Equations. American Journal of Computational Mathematics, 9, 251-260. https://doi.org/10.4236/ajcm.2019.94019