Research Article
An Improved Linearized Semi-implicit Runge-Kutta Methods
Department of Mathematics and Computer Science, Benson Idahosa University, Benin City, Edo State
Butchher (1963) gave the general R stage class of LSIRM for the initial value problem
(1) |
as
(2) |
Two-Stable Order 2 Methods (The 2-2 Methods)
The 2-2 methods is given as
(3) |
(4) |
Using (3), C1K1+C2K2 is now compared with |
Therefore, the NEQs for the above process are:
C1+C2 = 1 C1α1+C2α2+C2β21 = ½, With β21 = 0 |
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:
yn+1-yn = h(1/4K1+3/4K2)+O(h3) |
(5) |
(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.,
C1 = 0, C2 = 1, β21 = 0, which are seen to satisfy (5). | (5a) |
The Stability Function: μ22(λh)
(6) |
K1 = f+α1hfyK1 |
Apply to the scalar test equation, then:
K1 = λyn+α1hλK1, since fy(λyn) = λ |
K2(1-α1λh) = λ(yn+hβ21K1) |
(7) |
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),
(8) |
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:
• | it is A-Stable and |
• |
By LHospitals 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
• | α1>0; α2>0 and |
• | A12+A22 = B12+B22-2B2 |
where
A1 = 1-(α1+α2), A2 = α1α2+C1α2-C2α1+C2β21, B1 = α1+α2, B2 = α1α2 |
Proof
From the poles Lemma, the stability function μ22(Z) (8) will have poles at the points where B2Z2-B1Z+1 = 0
(9) |
The real roots are therefore = 1/α1, 1/α2 These will be positive only if 1/α1>0 and
1/α2>0 or α1>0 and α2 |
Also by the poles lemma, we expect |μ(it)|≤1 for all tεR.
(10) |
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
as Re(Z) →∞ |
By LHospitals Rule, since A1, B1 and B2 are constants. Therefore, A2 = 0 produces the required limit of μ22 as Re(Z) →-∞ but from (7)
A2 = α1α2+C1α2-C2 α1+C2β21 |
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
yn+1 = h(C1K1+C2K2)+O(h4) K12 = f2+2α1hf2fy+3 α12h2f2fy2+O(h3) |
(11) |
(12) |
(13) |
Similarly, B23 = fy(yn+hβ21K1)
(14) |
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),
(15) |
(16) |
From (12) also using (13), (15) and (16), Therefore,
(17) |
From (11)
(18) |
Comparing with
ΦT = f+h½ffy+h2(1/6f2fyy+1/6ff2y) |
Then, the NEQs are:
(19) |
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:
(20) |
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,
(21) |
From (17) | |||
K1 | = | f+hb21K1fy+h2/2b212K12fyy+α2hffy+h2α2β21K1ffyy+h2α2b21K1f2y+α22h2f2yK. Hence, | |
K2 | = | λyn+hb21K1λ+h2/2b212K12(0)+α2hλynλ+h2α2β21K1λyn(0)+h2α2b21K1λ2+α22h2λ2K2 |
(22) |
(23) |
(24) |
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.
C1 = -0.41, C2 = 1.41, α1 = 1.41, α2 = 0.59, b21 = 0.17, β21 = 0.17 |
A-Stability
A-Stability requires that
|μ22(Z)|<1 whenever Re(Z)<0 | |
i.e., | |1-Z-1.43Z2-2.18Z3-0.56Z4|<|1-2.34Z2+0.69Z4 |
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 LHospitals 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 eλh. 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
yn+1-yn = h(C1K1+C2K2+C3K3)+O(h)4 |
Then, | K1 = A33+α3hβ33(A33+α3hβ33K3) |
(25) |
(26) |
In view of the terms h and h2 in (26), we have:
(27) |
(28) |
Using K1 values in (26), we readily obtain:
In view of the terms α3h premultiplying A33B33 in (26) we have
A33B33 = ffy+hb31ff2y+hb32ff2y+hβ31f2fyy+hβ32f2fyy
(29) |
(30) |
So, we can write
(31) |
Furthermore we write
C1K1+C2K2+C3K3 = Cf+hN1ffy+h2N2f2fyy+h2N3ff2y |
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
and |
But K3 = A33+α3hA33B33+α32h2B332K3
Applying to the scalar test equation, we find that:
(1-α32(λh)2)K3 = T(A33+α2hA33B33) |
Table 1: | Global Error of y1 from yi* for each method at t = 0 |
or | (32) |
where T means applying the numerator to the scalar test equation.
We can readily obtain
This yields
(33) |
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.
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:
• | The R-stage-first order RO Method is the Euler scheme. | |
• | Theorem: A 2-2 RO is A-Stable if | |
• | α1 > 0, α2 > 0 and | |
• | A12+A22-2A2 = B12+B22-2B2 | |
• | Theorem: A condition for L-Stability of the 2-2 RO-Method is that |
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.