Received 15 December 2015; accepted 23 February 2016; published 26 February 2016
In this paper, we consider the first-order initial value problem of the form
with oscillatory/periodic solution.
Several classical methods (  -  ) for solving (1) have been derived. However, classical methods may not be well-suited for handling problems with pronounced periodic or oscillatory behaviour, because in order to accurately achieve this, a very small step size would be required with corresponding decrease in performance, especially in terms of efficiency. To overcome this barrier, classical methods have to be adapted in order to efficiently approach the oscillatory behaviour. The adaptation (which is called “exponential/trigonometric fitting”) is achieved by replacing some of the highest order monomials of the basis with exponentials or trigonometric (see  -  ). Numerical algorithms for solving problems whose solution exhibits a pronounced periodic or oscillatory behaviour has since the last decade gained a lot of attention. Such problems are often encountered in fields like mechanics, electronic, astrophysics, chemistry and engineering. The idea of using exponentially fitted formulae for differential equations was first proposed by Liniger and Willoughby  . Integration formulae containing free parameters were derived and these parameters were chosen so that a given function where q was real, satisfied the integration formulae exactly. This was tested on linear multistep method for, however Jackson and Kenue  derived a fourth order exponentially fitted formulae based on a linear 2-step formula and were A-stable. Based on this idea, Cash  , in his own work, attempted using Multiderivative Linear Multistep Method (MLMM) with k = 1 in the second derivative formulae. Particular Runge-Kutta (RK) algorithms have been proposed by several authors  -  in order to solve this class of problems. Vanden Berghe et al.   on the other hand, introduced other exponentially fitted RK (EFRK) methods which integrate exactly first-order systems whose solutions can
be expressed as linear combinations of functions of the form or.
Here, we analyze the construction and implementation of the exponentially-fitted variants of the 2-step Simpson method for solving problems of the form (1) which possess oscillatory/periodic solution, taking into account the six-step flow chart described by Ixaru and Vanden Berghe in  .
The main interest of this work is to modify the classical 2-step Simpson method for adaptation to oscillatory/ periodic problems.
2. Construction of Method
The classical 2-step Simpson method for solving (1) is given by
To begin the construction of the exponentially-fitted variants of (2), we rewrite (2) in a more general way as
Following the six-step flow chart, the corresponding linear difference operator reads
where. Applying step II of the six-step procedure, the resulting system of equations is compatible when. Solving the resulting system, we have
which are the coefficients of the classical method (2).
Applying step III, we find that
where and, (the frequency of oscillation) is real or imaginary. (For the trigonometric case, i.e., is imaginary, choose, i.e..)
To implement step IV, consider the reference set of M functions:
with. Since for our method, we have three possibilities:
•, the classical case with the set
•, the mixed case with the set
•, the mixed case with the set
The coefficients of the method for each case are obtained by the implementation of step V as follows:
S1: In this case, the solution is already known by (4)
As expected, the exponentially fitted variants reduce to the the classical method as.
3. Error Analysis: Local Truncation Error (lte)
The general expression of the leading term of the local truncation error (lte) for an exponentially fitted method with respect to the basis functions
takes the form (see  )
with K, P and M satisfying the condition.
For the three methods constructed above, one finds the following results:
4. Existence and Uniqueness of Solution
The following theorem states conditions of which guarantee the existence of a unique solution of the initial value problem (1)
Theorem 1. Let be defined and continuous for all points in the region defined by, , a and b finite, and let there exist a constant such that, for every such that and are both in,
then if is any given number, there exists a unique solution to the initial value problem (1), where is continuous and differentiable for all. Lambert  .
The requirement (12) is known as the Lipschitz Condition, and the constant is called the Lipschitz constant.
This condition may be thought of as being intermediate between differentiability and continuity, in the sense that
• continuously differentiable with respect to u
• Þ satisfies a Lipschitz Condition w.r.t. u
• Þ continuous w.r.t. u
In particular, if possesses a continuous derivative w.r.t. y for all, then, by the mean value theorem
where is a point in the interior of the interval whose end-points are u and, and and are both in. Clearly (12) is satisfied if
5. Contraction Mapping Theorem
In the sequel, we shall apply the following Contraction Mapping Theorem:
Theorem 2. (Contraction Mapping Theorem). Consider a set and a function. Assume
• D is closed (i.e., it contains all limit points of sequences in D)
• The mapping g is a contraction on D: There exists such that
• there exists a unique with
• for any, the fixed point iterates given by converges to as
• satisfies the a-priori extimate
and the a-posteriori error estimate
6. Application of the Contraction Mapping Theorem to LMM
If h is sufficiently small, implicit LMM methods also have unique solutions given h and. To see this, let be the Lipschitz constant for f. Given, the value for is obtained by solving the equation
That is, we are looking for a fixed point of
If, then is a contraction:
So by the Contraction Mapping Fixed Point Theorem, has a unique fixed point. Any initial guess yields a convergent fixed point iteration:
7. Convergence and Stability Analysis
Theorem 3 (Dahlquist Theorem) The necessary and sufficient conditions for a linear multistep method to be convergent are that it be consistent and zero-stable
Dahlquist theorem (3) holds also true for EF-based algorithms but, because their coefficients are no longer constants the concepts of consistency and stability have to be adapted.
Definition 4. An exponentially fitted method associated with the fitting space (9) is said to be of order, (where r is the order of the differential equation to be solved) and it is consistent if.
Since for all the constructed schemes, the consistency requirement is satisfied. Hence, the constructed schemes are all consistent.
Definition 5. A linear s-step method is said to be weakly stable if there is more than one simple root of the polynomial equation on the unit circle.
To investigate the stability of (3), one applies the method to the test problems. Applying (3) to the above test problems, one obtains
From the above, one finds that
where. The characteristics equation is given by
setting in (21), gives the reduced characteristic equation as. The roots are and hence the methods derived are weakly stable. Notice that depends on the test equation but Z on the numerical method.
Definition 6. A region of stability is a region of the q--z plane, throughout which. Any closed curve defined by is a stability boundary. Also, any interval of the real line is said to be the interval of stability if the method is stable for all
For each of the constructed methods, the region of stability is presented in Figure 1.
8. Numerical Results
Numerical experiments confirming the theoretical expectations regarding the constructed methods are now performed. The constructed methods are applied to two test problems and the result obtained compared with the classical fourth-order Taylor method, explicit four stage fourth-order Runge-Kutta method and the classical 2-step Simpson method.
8.1. Problem 1
Consider the IVP: with the exact solution. Solving the problem using different values of steplength h, the the maximum absolute errors for each steplength is obtained as presented in Figure 2. As expected, the exponentially-fitted variants (S2:(2,0), S3:(0,1)) of the classical 2-step Simpson method performed better compared with the classical methods.
8.2. Problem 2
Consider the IVP: with the exact solution. With, the problem is solved using different values of steplength h and the maximum absolute error for each steplength is obtained as presented in Figure 3. The constructed exponentially=fitted variants also performed better compared to the classical methods.
Figure 1. Truncated absolute stability regions of the constructed methods.
Figure 2. Maximum absolute errors for Problem 1 as a function of the step-size.
Figure 3. Maximum absolute errors for Problem 1 as a function of the step-size,.
The exponentially-fitted versions of the classical 2-step Simpson method have been constructed and imple- mented in this paper. The stability and convergence properties of the constructed methods were also analysed. The results obtained from the numerical examples show that the theoretical expectations are meet (i.e. the expo- nentially-fitted variants of the classical 2-step Simpson method are suitable for solving periodic/oscillatory problems).
We thank the Editor and the referee for their comments.
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