Abstract
We derive a new class of linear multistep methods (LMMs) via the interpolation and collocation technique. We discuss the use of these methods as boundary value methods and block unification methods for the numerical approximation of the general second-order initial and boundary value problems. The convergence of these families of methods is also established. Several test problems are given to show a computational comparison of these methods in terms of accuracy and the computational efficiency.
1. Introduction
Linear multistep methods (LMMs) are widely used for the numerical integration of ordinary differential equations. They are a class of -step difference equations of the formwhere , are coefficients to be uniquely determined, is the order of the differential equation whose solution is being sought, is the constant stepsize, , and .
To be able to use (1), we need to impose additional conditions. Initial value methods (IVMs) are methods whose additional conditions are specified as initial conditions so that they form discrete initial value problems. The IVMs are used for the numerical integration of initial value problems [1–4]. However, if these additional conditions are specified as initial and final conditions (or methods) so that they form a discrete analog of the continuous boundary value problems, we have the boundary value methods (BVMs). They are used for the approximation of both initial and boundary value problems [5–11]. The BVMs are a larger class of methods that contains the IVMs since the IVMs are BVMs with zero final conditions. Sometimes the additional conditions are given as a set of LMMs which together with the main method (1) forms the block methods. If the union of the methods in the block is obtained for , where is the number of grid points, so that we have difference equations in unknowns (grid values) which can be easily solved, the resulting approach is termed the block unification methods (BUMs) [12]. The union of the methods in the block is taken to have a consistent equation.
In what follows, we will consider the general second-order system of the formcoupled with any of the initial or boundary conditionswhere are continuous functions. The existence and uniqueness of the solutions of (2) subject to any of (3) have been given in Wend [13] or Ascher et al. [14].
The BVMs have been used for the numerical integration of first-order initial and boundary value problems and their convergence and stability analysis have been fully discussed [5–10]. Aceto et al. [15] constructed P-stable LMMs which were used as BVMs for the special second-order problem . Recently, Biala and Jator [11] developed BVMs for the direct solution of the general second-order initial and boundary value problems arising from the semidiscretization of three-dimensional partial differential equations.
The BUMs have also been successfully applied to solve initial and boundary value problems [12]. In this paper, we construct a new class of LMMs which we implement as boundary value methods and block unification methods. We compare the results of these two classes of methods in terms of accuracy and CPU time.
The outline of the paper is as follows: In Section 2, we derive a -step continuous LMM (CLMM) via the interpolation and collocation technique [2–4, 11, 12]. In Section 3, we construct the BVMs by using the CLMM to derive a -step discrete LMM which is to be used with some initial and final methods (also obtained from the CLMM). We also discuss the convergence and the use of the BVMs in this section. Section 4 details the BUMs. Their convergence analysis is carried out and an algorithm for their implementation is also discussed. In Section 5, we give several numerical test problems which were solved using both the BVMs and the BUMs. Their comparison in terms of accuracy and computational efficiency (CPU Time) was also shown. Finally, we give some concluding remarks on the methods in Section 6.
2. Derivation of the CLMM
In this section, we will construct a -step CLMM using the interpolation and collocation technique. The CLMM will be used to generate the BVMs and the BUMs.
We begin by constructing the CLMM of the formwhere , , , and are continuous coefficients. The next theorem discusses the construction of the CLMM.
Theorem 1. Let (4) satisfy the following equations:then the continuous representation (4) is equivalent towhere ; are basis functions and the matrix is defined as follows: is obtained by replacing the jth column of by , wherewhere denotes the transpose.
Proof. We begin the proof by assuming polynomial basis functions of the formwhere , are coefficients to be determined.
By substituting (9) into (4), we havewhich is simplified toand expressed in the formwhereImposing conditions (5) on (12), we obtain a system of equations which can be expressed as , where is a vector of undetermined coefficients.
Using Cramer’s rule, the elements of are determined and given aswhere is obtained by replacing the th column of by . We rewrite (12) as (6) using the newly found elements of .
Remark 2. It has been shown in [11] that symmetric schemes are the best candidates to be used as final methods. Thus, CLMM (4) is chosen to ensure that we have discrete symmteric schemes by evaluation at some points .
3. The Boundary Value Methods
CLMM (4) is evaluated at , , , to obtain the BVMs. The main method of the BVM, that is, , is of the formwhose derivative formula, obtained by evaluating at , is
3.1. Convergence of the BVMs
In this section, we will discuss the convergence of the BVMs. We emphasize that (4) is evaluated at , , , to obtainand also, by evaluating at , , we obtain the derivative formulas
We note that the formulas in (17) and (18) are of . We establish the convergence of the BVMs in the following theorem.
Theorem 3. Let be an approximation of the solution vector for the system obtained on a partition from methods (17) and (18). If , , where the exact solution is several times differentiable on , and if , then the BVM is convergent and of order , which implies that .
Proof. We compactly write (17) and (18) in matrix form by introducing the following matrix notations. Let be a matrix defined bywhere are matrices given as is an null matrix, and is an identity matrix.
Similarly, let be a matrix defined bywhere are matrices given asand , are null matrices.
We also define the following vectors:The exact form of the system formed by (17) and (18) is given bywhere is the truncation error vector of the formulas in (17) and (18). The approximate form of the system is given bywhere is the approximate solution of vector .
Subtracting (24) from (25) and letting and using the mean value theorem, we have the error systemwhere is the Jacobian matrix and its entries , , , and are defined asLet be a matrix of dimension so that (26) becomesand, for sufficiently small , is a monotone matrix and thus nonsingular (see [16]). Thereforewhich shows that the methods are convergent and the global error is of order .
3.2. Use of Methods
The BVMs can only be successfully implemented if used together with appropriate additional methods [5]. In this regard, we have proposed a main method and additional methods which are obtained from the same continuous scheme (the CLMM).
To use LMM (15) as BVMs, we rewrite main method (15) aswith the derivative formulawhich are to be used with some boundary conditions and , .
The discrete solutionsare to be obtained for methods (30) and (31) to be useful. However, (3) provides two solution values so that we have to impose additional conditions of which , , gives methods and the remaining methods are given as a set of initial and final methods which is readily obtained from CLMM (4). Approximations (32) need be at least of order accuracy if BVM (30) is of order in order to have a solution accuracy of . Equations (30) and (31) with the additional methods give a set of equations in unknowns which can be easily solved. We give below the BVMs of orders 6 and 8.
BVM of Order 6 with the derivative formulaswhich are to be used with the initial method and the final method
BVM of Order 8 with the derivative formulaswhich are to be used with the initial methodsand the final methods
4. The Block Unification Methods
The BUMs are also a class of methods for the numerical integration of both initial and boundary value problems. CLMM (4) is a -step continuous scheme and as such the BUM requires a set of methods so that, on the partition , , , the solution of the step is obtained. CLMM (4) is used to generate methods by evaluating (4) at and also evaluating at . CLMM (4) can only be used to construct methods and as such we give the last method as (since it is also of )The BUM ( methods) is of the formFormulas (42) that form the BUM are all weighted the same unlike the BVMs that have main methods (15) and (16). We give below the BUMs of orders 6 and 8.
BUM of Order 6
BUM of Order 8
4.1. Convergence and Use of the BUMs
BUMs (42) are weighted the same as each formula in (42) is used the same number of times as others. Their convergence was established in a similar way to Theorem 3 with some changes in the coefficients of the matrices and the global error is also of .
The BUM is implemented efficiently by using the following algorithm.
Step 1. Use the block unification of (42) for to obtain in the interval ; for , is obtained in the interval ; and in the intervals for , we obtain where .
Step 2. The unified block given by the system obtained in Step 1 results in a system of equations in unknowns which can be easily solved.
Step 3. The values of the solution and the first derivatives of (2) are generated by the sequence of , , , obtained as the solution in Step 2.
5. Test Problems
We consider five numerical examples. The examples were solved using the BVMs and the BUMs of different order derived in this paper. Comparisons are made between the BVMs and BUMs by obtaining the maximum errors in the interval of integration. We note that the number of function evaluations (NFEs) involved in implementing the two methods is in the entire range of integration. In order to show the competitiveness of the derived methods with some existing methods in the literature, we compared our methods with the Extended Trapezoidal Rules (ETRs), Extended Trapezoidal Rules of the second kind (ETR2s), and the Top Order Methods (TOMs) of orders 6, 8, and 10, respectively, given in [6]. For linear problems, we solve the resulting system of equations using Gaussian elimination with partial pivoting and, for nonlinear problems, we use a modified Newton-Raphson method.
Example 1. We consider the boundary value problem given in [6]:
Example 2. We consider the nonlinear Fehlberg problem given in [12]:
Example 3. We consider the nonlinear BVP with mixed boundary conditions given in [17]:
Example 4. We consider the nonlinear BVP given in [18]:where
Example 5. Lastly, we consider the following BVP for given in [19]:
5.1. Numerical Results and Discussion
Example 1 is a variable coefficient fourth-order BVP. The fourth-order BVP is transformed to a system of second-order BVP. We solved the system using the BVM and BUM of orders 6 and 8. The problem is also solved using the ETRs and ETR2s of orders 6 and 8, respectively. Tables 1 and 2 show the computational results for this example. While the BUM produces solutions of approximate accuracy with the BVM, it uses shorter CPU Time. Example 2 is the well-known nonlinear Fehlberg problem. It was solved for and the maximum of the Euclidean norm of the errors in and was obtained in the interval of integration. Example 2 was also solved using the ETR2s and the TOMs of orders 8 and 10, respectively. Tables 3 and 4 show that both methods produce solutions of approximate accuracy with the BUM using shorter CPU Time. Example 3 was chosen to demonstrate the use of the methods on a nonlinear BVP with mixed boundary conditions. The computational results were given in Tables 5 and 6. Example 4 was chosen to show the performance of the schemes on systems of nonlinear BVPs. The maximum of the Euclidean norm of the errors in and is given in Tables 7 and 8. Lastly, we show the performance of the methods on a Poisson equation with boundary conditions. The partial differential equation is transformed into a system of second-order ordinary differential equations with boundary conditions using the method of lines. Table 9 shows the computational result for this example. Also, Figures 1–4 show the efficiency curves of these methods for the different examples where we have denoted BVM and BUM with as BVM2 and BUM2, respectively.
From the foregoing, it can be concluded that the BUM and the BVM produce solutions of approximate accuracy with the BUM using shorter CPU Time. However, a -step BVM performs poorly when the number of steps, , is . This is because the main method together with the initial and final methods does not form a good discrete analog or approximation of the continuous boundary value problem. Also, the BUM has the drawback that it is only implemented for any which is a multiple of .
6. Conclusion
In this paper, we have developed a new class of LMMs and implemented the LMMs via two approaches, the boundary value approach and the block unification strategy, which are used to solve initial and boundary value problems. The comparison of the two approaches was carried out in terms of accuracy and computational efficiency. The results given in Section 5 show that both approaches perform very well with the BUM using shorter CPU Time. Our future research will be to develop a variable stepsize version of the BVM and the BUM and a study of the conditioning of the matrices arising from the discretization of the continuous second-order problems.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.