Implementing a seventh-order linear multistep method in a predictor-corrector mode or block mode: which is more efficient for the general second order initial value problem
© Jator and Lee; licensee Springer. 2014
Received: 14 July 2014
Accepted: 12 August 2014
Published: 20 August 2014
A Seventh-Order Linear Multistep Method (SOLMM) is developed and implemented in both predictor-corrector mode and block mode. The two approaches are compared by measuring their total number of function evaluations and CPU times. The stability property of the method is examined. This SOLMM is also compared with existing methods in the literature using standard numerical examples.
AMS Subject Classification
Some of the methods available for directly solving (3) are due to Awoyemi (2001) and Ramos and Vigo-Aguiar (2006). These methods are generally implemented in a step-by-step fashion in a predictor-corrector mode.
In this paper, we construct the continuous form of (1) which has ability to generate several methods which are combined and implemented in block form to solve (3) directly (see Jator and Li (2009) and Jator (2012, 2010, 2007).
The paper is organized as follows. In Section ‘SOLMM’, we derive a continuous approximation which is used to obtain the discrete methods that are combined to form the block method. The analysis and computational aspects of the SOLMM is given in Section ‘Implementation of the SOLMM’. Numerical examples are given in Section ‘Numerical examples’ to show the accuracy and efficiency of the method. Finally, the conclusion of the paper is discussed in Section ‘Conclusion’.
where α0(t), α1(t), and β j (t), j = 0,1,2 are continuous coefficients that are uniquely determined. We assume that yn + j is the numerical approximation to the analytical solution y(tn + j), y n + j′ is an approximation to y′(tn + j), and fn + j = f(tn + j,yn + j,y n + j′), j = 0,1,…,6 is supplied by the differential equation. The coefficients of the method (4) are specified by the following theorem.
and W j is obtained by replacing the j t h column of W by V; P j (t) = t j ,j = 0,…,8 are basis functions, and V is a vector given by V = (y n ,yn + 1,f n ,fn + 1,…,fn + 6) T . We note that T is the transpose.
See Jator (2012).
and A0, A1, B0, and B1 are matrices of dimension 12 whose entries denoted by α j = αi,j, β j = βi,j,i = 1,…,12 are given by the coefficients of (8) and (9).
Order and local truncation error
and Ł[ z(t);h] = (Ł1[ z(t);h],…,Ł6[ z(t);h],Ł7[h z′(t);h],…,Ł12[h z′(t);h]) T is a linear difference operator.
where ∥·∥ is the maximum norm.
The block method (10) is said to be consistent if it has order at least one.
The block method (10) has order and error constant given by the vector p=6 and .
Linear stability of the SOLMM
where the matrix M(q) is the amplification matrix which determines the stability of the method.
The interval [ -q0,0] is the stability interval, if in this interval ρ(q)≤1, where ρ(q) is the spectral radius of M(q) and q0 is the stability boundary (see Sommeijer (1993)).
We found that ρ(q)≤1 if q ε [-4.552,0], hence, for the SOLMM, q0 = 4.552.
Implementation of the SOLMM
Results, with t ε [ 0,1], for Example 1
Results, with t ε [1,8], for Example 2
Results, with , for Example 3
Predictor-corrector mode algorithm
The initial block was used to start predictor-corrector algorithm, after which the predictor (14) and corrector (15) were used in a step-by-step fashion to provide the numerical solution from the second block to the end of the interval.
The theoretical solution at t=8 is .
Comparison of block mode and predictor-corrector mode
The SOLMM is implemented in both predictor-corrector and block modes. The two approaches are compared by measuring their total number of function evaluations (NFEs) and CPU times in seconds. The block mode implementation is shown to be superior to the predictor-corrector mode implementation in terms of accuracy and the number of function evaluations. However, the predictor-corrector mode implementation uses less time than the block implementation. Details of the numerical examples are displayed in Tables 1, 2 and 3.
Comparison of block method with other methods
Absolute errors for Example 2
The correct decimal digit at the endpoint for Example 3
A SOLMM is proposed and implemented in both predictor-corrector and block modes. It is shown that the block mode algorithm is superior to the predictor-corrector mode algorithm in terms of accuracy and the number of function evaluations. However, the predictor-corrector mode implementation uses less time that the block implementation. the Details of the comparison of the numerical examples are displayed in Tables 1, 2, 3, 4 and 5. Our future research will be focus on developing a variable step version of the SOLMM in both modes.
- Awoyemi DO: A new sixth-order algorithm for general second order ordinary differential equation. Int J Comput Math 2001, 77: 117-124. 10.1080/00207160108805054View ArticleGoogle Scholar
- Hairer E: Méthodes de Nyström pour l’équation différentielle y′′ = f ( x , y ). Numer Math 1977, 25: 283-300.Google Scholar
- Ixaru L, Berghe GV: Exponential fitting. Kluwer, Dordrecht, Netherlands; 2004.View ArticleGoogle Scholar
- Jator SN: A continuous two-step method of order 8 with a Block Extension for y′′ = f ( x , y , y′). Appl Math Comput 2012, 219: 781-791. 10.1016/j.amc.2012.06.027View ArticleGoogle Scholar
- Jator SN: Solving second order initial value problems by a hybrid multistep method without predictors. Appl Math Comput 2010, 217: 4036-4046. 10.1016/j.amc.2010.10.010View ArticleGoogle Scholar
- Jator SN, Li J: A self-starting linear multistep method for a direct solution of the general second order initial value problem. Intern J Comput Math 2009, 86: 827-836. 10.1080/00207160701708250View ArticleGoogle Scholar
- Jator SN: A sixth order linear multistep method for the direct Solution of y′′ = f ( x , y , y′). Intern J Pure Appl Math 2007, 40: 457-472.Google Scholar
- Lambert JD, Watson A: Symmetric multistep method for periodic initial value problem. J Instit Math Appl 1976, 18: 189-202. 10.1093/imamat/18.2.189View ArticleGoogle Scholar
- Ramos H, Vigo-Aguiar J: Variable stepsize Stôrmer-Cowell methods. Math Comp Mod 2005, 42: 837-846. 10.1016/j.mcm.2005.09.011View ArticleGoogle Scholar
- Sommeijer BP: Explicit, high-order Runge-Kutta-Nyström methods for parallel computers. Appl Numer Math 1993, 13: 221-240. 10.1016/0168-9274(93)90145-HView ArticleGoogle Scholar
- Vigo-Aguiar J, Ramos H: Variable stepsize implementation of multistep methods for y′′ = f ( x , y , y′). J Comput Appl Math 2006, 192: 114-131. 10.1016/j.cam.2005.04.043View ArticleGoogle Scholar
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