Trends in Applied Sciences Research

Year: 2006 | Volume: 1 | Issue: 5 | Page No.: 444-455
DOI: 10.17311/tasr.2006.444.455

An Improved Linearized Semi-implicit Runge-Kutta Methods

F.E. Bazuaye

Abstract: We present a framework for the stability analysis of a linearized semi-implicit Runge-Kutta Method. To determine an A or L stable LSIRM. The conventional procedure has been to estimate the parameters from the construction and then test for A-or L-stability. It is now questioned whether conditions cannot be imposed on the parameters a priori to achieve the required stability. The general LSIRM of order two and three respectively were, therefore, constructed using Taylor’s expansion. The stability function for each method was derived from the stability analysis; criteria were set for A-or L-stability.

Keywords: complex plane, Linearized methods, stability function and rosenbrock

Introduction

Butchher (1963) gave the general R stage class of LSIRM for the initial value problem

as

Two-Stable Order 2 Methods (The 2-2 Methods)

The 2-2 methods is given as

Therefore, the NEQs for the above process are:

We have the equations in five unknowns. Fatunla (1988) and Iserles (1996). Thus, we are free to fix three parameters giving a three parameters family of solutions. For example, let: C₁ = ¼, α₁ = ½, β₂₁ = -¹/_6,

Then, solving the NEQs, we have:
C₂ = ¾, α₂ = ²/₃, β = 0. The resulting method is:

(The above method is L-Stable and will be seen from theorem 2.

Rosenbrock (1963) gave the following values for the parameters of a 2-2 method, i.e.,

The Stability Function: μ₂₂(λh)

Apply to the scalar test equation, then:

Stability Analysis
The method given by Rosenbrock (1963) and Okonta (2004) is said to be A-Stable. We prove that this is true.
For the method

Substituting into (7),

A-Stability
By definition, the method is A-Stable if |μ₂₂(Z)|<1 whenever Re(Z) <0. This reduces to

Take a test case of Re(Z) = -1
Then

|0.5859|<|1.6715|which is true. Hence, the method is confirmed to be A-Stable.

L-Stability
By definition, the method is L-Stable if:

By L’Hospital’s Rule, the method is stable.

Poles Lemma
Let r be an arbitrary rational function which is not a constant. Then, |r(Z)|< 1 for all ZεC, where C is the complex plane, if and only if all the poles of r have positive real parts and |r(it)≤1 for all tεR.

Theorem 2.2
A two stage 2nd order LSIRM is A-stable if

where

Proof
From the poles Lemma, the stability function μ₂₂(Z) (8) will have poles at the points where B₂Z²-B₁Z+1 = 0

The real roots are therefore = 1/α₁, 1/α₂ These will be positive only if 1/α₁>0 and

Also by the poles lemma, we expect |μ(it)|≤1 for all tεR.

If the parameters of the method are not given then, A₁, A₂, B₁, B₂ may be found from the stability function.

, It is easily seen from μ₂₂(Z) that

, Which satisfies condition (i) of the theorem.

Also, from

obtained by application of (5).
It is seen that

which altogether prove that the method is A-Stable.

Having known the conditions on the parameters that will lead to A-Stability, it becomes necessary to examine the conditions on the parameters that may guarantee L-Stability.

This leads us to propose the following theorem.

Theorem
Given the parameters α₁, α₂, C₁, C₂ and β₂₁ in a 2-stage second order Rosenbrock method, a condition for L-stability is that:

Proof
The stability function is

Since β₂>0 by theorem 2.1 putting A₂ = 0 and find the limit of μ₂₂(Z) as Re(Z) →–∞

The limit

By L’Hospitals Rule, since A₁, B₁ and B₂ are constants. Therefore, A₂ = 0 produces the required limit of μ₂₂ as Re(Z) →-∞ but from (7)

Therefore, if A₂ = 0

(Verification shows that (3.2) is L-stable)
The parameter of the method is μ₂₂(Z).

Two-Stage, Order 3 Methods (The 2-3 Method)
Construction

Similarly, B₂₃ = f_y(y_n+hβ₂₁K₁)

In (12) we shall need A₂₃, B₂₃ and B₂₃². By inspection of (12), we observe that the factors α₂h and α₂²h² permit us to discard O (h²) in the expansion of A₂₃, B₂₃ and discard O(h) in the expansion of B₂₃² and K₂.

Hence, using (13) and (14),

From (12) also using (13), (15) and (16), Therefore,

From (11)

Comparing with

Then, the NEQs are:

The exact replica of these equations is found in Reosenbrock (1963) and Lambert (1977). The following values of the parameters are in perfect agreement with (12) and are said to produce an A-stable 2-3 Method.

These are:

The set (20) has four equations in six unknowns. So, we have 2 free parameters producing a 2 parameter family of solution.

Stability Function μ₂₂(λh)

Applying to the scalar test equation,

The method (23) from Lambert (1977), is to be investigated for the purpose of demonstration we approximate the given values of the parameters to two decimal places.

A-Stability
A-Stability requires that

For any Re(Z) < 0

Taken the test case, Re(Z) = -1. Then,

|0.19|<|0.65|

0.19 < 0.65, which is true. For Re(Z) =-2, |1.76|<|2.68|, etc.
Hence the method is A-Stable as asserted by Lambert (1977).

L-Stability
The requirement is that

Lt |μ₂₃(Z)|→0 as Re(Z)→-∞

By L’Hospitals rule, this reduces to 0.8 as R e(Z)→-∞

This is exactly the same limit obtained by Rosenbrock (1963), for the same method using the function ψ(t) defined in the literature as an approximation to e^λh. Since it |μ₂₃(Z)|≠0 as Re(Z)→-∞. Hence, the method is not L-stable

Three Stage Order 3 Methods (The 3-3 Methods) Construction

In view of the terms h and h2 in (26), we have:

Using K₁ values in (26), we readily obtain:

In view of the terms α₃h premultiplying A₃₃B₃₃ in (26) we have

A₃₃B₃₃ = ff_y+hb₃₁ff²_y+hb₃₂ff²_y+hβ₃₁f²f_yy+hβ₃₂f²f_yy

So, we can write

Furthermore we write

Comparing with

Where the normal equations now become:

These are four equations in 1 unknowns which yield 8 free parameters and can therefore have 8 parameter family of solutions.

Stability Function μ₃₃(λh)

Applying K₁ and K₂ of order 3 to the test equation, we obtain

But K₃ = A₃₃+α₃hA₃₃B₃₃+α₃²h²B₃₃²K₃

Applying to the scalar test equation, we find that:

where T means applying the numerator to the scalar test equation.

We can readily obtain

This yields

Numerical Experiments

In this section, we apply the methods discussed in this paper to a particular differential system on a comparative basis.

Problem

Using a step size h = 0.1, find the approximate solution yi of the system at t = 0.1 using:
The 2-stage, 2nd order L-Stable Rosenbrock Method.
The 2-stage, 3rd order A-Stable Rosenbrock Method.
The linear multi-step implicit Euler scheme of order.

Solving the above problems with the methods discussed in this study, we obtained the result as shown in Table 1.

Results and Discussion

In this study, we constructed the general Rosenbrock methods of orders two and three respectively, using Taylors expansion. The stability function for each method was derived from the stability analysis or application of each stability function; criteria were set for A or L-stability. Some of these criteria involving the parameters of the methods were established.

The following major results and theorems were obtained:

Conclusions

From the conditions imposed on the parameters of the 2-2 and 2-3 RO-Methods for A-or L-Stability, the need for this study has been substantially realized.

Armed with these criteria, we will be able to know in advance whether a set of parameters selected for the said methods will be A-or L-stable. This is a fair departure from the tradition of first determining the parameters and, then testing for A-or L-stability.

REFERENCES