Diagonally implicit Runge-Kutta fourth order four-stage method for linear ordinary differential equations with minimized error norm

We constructed a new fourth order four-stage diagonally implicit Runge-Kutta (DIRK) method which is specially designed for the integrations of linear ordinary differential equations (LODEs). The method is obtained based on the Butcher’s error equations. In the derivation, the error norm is minimized so that the free parameters chosen are obtained from the minimized error norm. Row simplifying assumption is also used so that the number of equations for the method can be reduced and simplified. A set of test problems are used to validate the method and numerical results show that the new method is more efficient in terms of accuracy compared to the existing method.


Introduction
We consider the numerical integration of linear inhomogeneous systems of ordinary differential equations (ODEs) of the form where A is a square matrix whose entries does not depend on y or x , and y and ) (x G are vectors.Such systems arise in the numerical solution of partial differential equations (PDEs) governing wave and heat phenomena after application of a spatial discretization such as finite-difference method.
To derive Runge-Kutta (RK) methods, we need to fulfill certain order equations; see Dormand [3].These order equations resulted from the derivatives of the function ) , ( y x f y = ′ itself.If the function is linear then some of the error equations resulted by the nonlinearity in the derivative function can be removed, thus less order equations need to be satisfied, hence a more efficient method in some respect than the classical method can be derived.
In this paper, we construct diagonally implicit Runge-Kutta method specifically for linear ODEs with constant coefficients.We consider the principal terms of the local truncation error to minimize the error norm.Then, the stability aspect of the method is looked into and a few test equations are used to validate the new method.

Derivation of the Method
In this section, we consider the following scalar ODE ) , ( y When a general s-stage diagonally implicit Runge-Kutta method is applied to the ODE, the following equations are obtained, (2.2a) where We shall always assume that the row-sum condition holds .According to Dormand [3], the following eight order equations (error equations) are equations needed to be satisfied by fourth order four-stage DIRK method.
The restriction to linear ODEs reduces the number of equations which the coefficients of the RK method must satisfy in table 2.1.Zingg and Chisholm [8] have derived new explicit RK methods which are suitable for linear ODEs that are more efficient than the conventional RK methods.
For this new fourth order DIRK method which is suitable for linear ODEs, equation 6 in table 2.1 can be eliminated, as in [8].This condition is eliminated by exploiting the fact that, for linear ODEs, Using the simplifying assumption: we can removed several equations, i.e. equation 4 and 7 in table 2.1.This makes the new fourth order four-stage DIRK method different from the classical method.So, the number of order equations can be reduced.Thus, the equations needed to be satisfied are: Altogether there are seven equations to be satisfied and we have 10 unknowns.So, we can take three free parameters which are chosen to be    τ .The best strategy for practical purposes would be to choose the free RK parameters is to minimize the error norm, see [3]; So we have the principal error norm for this method; τ is the error equations associated with the fifth order method, (in table 2.2).Then we get the principal error norm in terms of

Stability
One of the practical criteria for a good method to be useful is that it must have region of absolute stability.When an s-stage Runge-Kutta method (equations (2.2a) and (2.We can solve for h ˆ using the Mathematica packaged and get the stability polynomial and also the stability region.The stability polynomial for new fourth order four-stage DIRK method is ( )

h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h
The stability polynomial is set to zero and solve for h ˆ which gives the value of 1 ) ( ≤ h R ; this is done by using Mathematica package.The stability region is obtained by tracing the values of h ˆ and is shown in Figure 2.3.The stability region for new fourth order four-stage DIRK is black in colour.

Results and Discussion
The following are some of the problems tested.All the problems are linear ODEs.The numerical results are tabulated and compared with the existing method and below are the notations used:

Conclusion
The new fourth order four-stage DIRK method with minimized error norm has been presented for the integration of linear ODEs.It has a bigger stability region compared to explicit RK method (of the same order), hence more stable.From the numerical results in Table 3.1 to 3.3, we can conclude that the new fourth order four-stage DIRK method which is suitable for linear ODEs performs better in terms of maximum error compared to fourth order four-stage ERK method [8].This new method is also as good as the optimal fourth order fourstage singly-DIRK [4].

τΓ
In order to choose the free parameters 3 2 , c c and γ , the principal terms of the local truncation error must be considered.Using the error function For case of RK suitable for linear ODEs, we only considered .Here we can eliminated several equations i.e.
and γ and solving all the equations we finally get all the coefficients as follows; is (m x m), e is (m x 1) and ) (h R is called the stability polynomial of the method.The stability region is
Substituting all the parameters into the general form of RK method, we have the new fourth order four-stage DIRK method which is suitable for linear ODEs with minimized error norm,