- Research
- Open Access
Dynamical analysis in explicit continuous iteration algorithm and its applications
- Qingyi Zhan^{1}Email authorView ORCID ID profile,
- Zhifang Zhang^{2}Email author and
- Xiangdong Xie^{3}
https://doi.org/10.1186/s13662-018-1909-z
© The Author(s) 2018
- Received: 17 July 2018
- Accepted: 2 December 2018
- Published: 11 December 2018
Abstract
This article is devoted to the dynamical analysis of an explicit continuous iteration algorithm, describing its construction, relationship with the explicit trapezoid method, and error analysis. A theorem demonstrating the equality of these methods is also established. The accuracy of the theoretical results and universality of the explicit continuous iteration algorithm are proved by numerical experiments.
Keywords
- Error analysis
- Explicit continuous iteration algorithm
- Implicit method
- Numerical simulation
1 Introduction
With developments in the society and economy, scientific computing has recently become increasingly popular in the world. In fact, it is essential that we derive high order, efficient numerical methods to solve differential equations, which are widely used in physical problems. In particular, it is very important to construct fast algorithms for solving practical problems.
It is well known that many numerical methods are applied to mathematical models to investigate the solution space. Some of these methods are explicit, such as Euler scheme, Adams scheme, and Runge–Kutta scheme, we refer the reader to [9, 10] and the references therein. Others employ implicit methods. However, implicit approaches have many shortcomings, such as being overly complex, relatively slow, and requiring excessive internal memory space. Therefore, explicit methods have become more widely used.
Many uncertainties and practical difficulties involved in models for solving differential equations mean that there are relatively few reports in the literature up to now. In [1], Butcher stated that the classic finite stage Runge–Kutta methods could be expanded to infinite stage Runge–Kutta methods, and suggested that the finite summation should be changed to definite integration over finite intervals. However, he did not make any further progress in this field. In 2010, Haier built on this important concept and provided an expression for a continuous stage Runge–Kutta method [2]. We expand this to the case of ordinary differential equations (ODEs), which describe many natural phenomena in meteorology, biology, and so on [7, 8]. To the best of our knowledge, there are no previous reports of explicit continuous iterative methods in the literature.
The main motivations for this work are twofold. On the one hand, the classical results on explicit numerical methods are the basis for this research. A variety of numerical methods have been applied to different aspects of differential equations, and many important results have revealed the mechanisms of dynamical behavior. On the other hand, our earlier work [10, 11] on stability analysis and numerical simulations of stochastic differential equations have inspired further study in this direction. For example, there has been some research on the numerical analysis [5, 6, 11] and numerical simulations [10] of stochastic differential equations. These studies established the foundation of numerical analysis.
In this study, we first construct a class of explicit continuous iterative (ECI) algorithms, and then compare with other classes in terms of class of numerical methods and error analysis. Numerical examples are presented to illustrate the feasibility of the ECI algorithm and to provide accurate solutions within a reasonable time. These results show that, under some appropriate conditions, ECI algorithm can be used to solve some nonlinear ODEs more accurately than with some existing numerical approximations.
The remainder of this paper is organized as follows. Section 2 describes the construction of the ECI algorithm, and introduces some relevant concepts and norms which will be utilized later. Section 3 is devoted to the theoretical analysis of the ECI algorithm, i.e., the error analysis of the solution and equivalence properties. Section 4 presents numerical experiments in some given areas, including illustrative numerical results for the main theorem. Section 5 provides the conclusions to this study.
2 Construction of explicit continuous iterative algorithm
Theorem 2.1
\(Y_{n}\) and \(Y_{n+1}\) obtained by scheme (1) are the same as those by the trapezoid formula.
Proof
Remark 1
As known from the above results, the essence of the ECI algorithm is that explicit iteration is applied to fast obtain approximate solutions, which are continuous and more accurate, to the true solutions.
3 Error analysis
Lemma 3.1
Proof
Secondly, we have \(U(nh)=Y_{n}\).
Lemma 3.2
Proof
This completes our proof. □
By the conclusions of Lemmas 3.1 and 3.2, we obtain the following error control theorem.
Theorem 3.1
Proof
Therefore, the conclusion of Theorem 3.1 follows from Lemma 3.1. □
Remark 2
The advantages of this method are not only in its convergence, that is, the iterated error being limited in a small interval, but also in its ability to simulate the solutions of ODEs continuously and explicitly, which can help simulate the true solutions more accurately.
4 Numerical experiments
4.1 Comparison with classic methods
As for the test equation in Sect. 2, we only consider the special case \(Y\in \mathbb{R}\), \(a=-4.0\) and \(Y(0)=1.0\). We compare numerical solutions obtained by the ECI algorithm with those yielded by some classic methods, such as Euler method and implicit trapezoid method. We choose the step size \(h=0.01\), and the different results are shown as follows.
Comparison of numerical solutions for different number of steps, N, obtained by different methods
N | Euler method | ECI algorithm | Trapezoid method |
---|---|---|---|
0 | 1.0 | 1.0 | 1.0 |
50 | 0.1353 | 0.1408 | 0.1408 |
100 | 0.0176 | 0.0191 | 0.0191 |
200 | 2.9647e − 04 | 3.4878e − 04 | 3.4878e − 04 |
500 | 1.4235e − 09 | 2.1396e − 09 | 2.1396e − 09 |
1000 | 1.9452e − 18 | 4.3982e − 18 | 4.3982e − 18 |
Comparison of errors for different number of steps, N, obtained by different methods
N | Euler method | ECI algorithm | Trapezoid method |
---|---|---|---|
0 | 0 | 0 | 0 |
50 | 3.7582e − 05 | 0.0055 | 0.0055 |
100 | 7.4239e − 04 | 7.3741e − 04 | 7.3741e − 04 |
200 | 3.8996e − 05 | 1.3320e − 05 | 1.3320e − 05 |
500 | 6.3769e − 10 | 7.8414e − 11 | 7.8414e − 11 |
1000 | 2.3032e − 18 | 1.4988e − 19 | 1.4988e − 19 |
4.2 Applications in numerical simulations
And we obtain the computational time when the error tolerance is no more than \(\|E\|=4.91\mbox{e}{-}06\). The results are as follows.
Comparison of various characteristics for different methods
Main properties | Euler method | ECI algorithm | Trapezoid method |
---|---|---|---|
computational time | 4.431 | 3.852 | 4.231 |
number of iterations | 890 | 773 | 1369 |
average iteration times | 2.011 | 2.008 | 3.237 |
used CPU time | 0.527 | 0.471 | 0.508 |
Remark 3
As we see from this numerical experiment, although the ECI algorithm is constructed for simple test equations, it can be extended to some nonlinear ODEs, which can generate dynamical systems by some parameter transformations. However, the conditions, which should be satisfied for such nonlinear ODEs, are still to be investigated, and the associated algorithm will be revised, if needed. All these questions will be tackled in our future work.
5 Conclusion
The main result of this paper is the dynamical analysis of the ECI algorithm and its applications in simulating the solutions of ODEs. The results show that this algorithm is effective and the numerical results can match the results of theoretical analysis. Although some progress is made, more practical models and methods, which are needed to solve a system of ODEs or stochastic differential equations, will be shown in our future work.
Declarations
Acknowledgements
The authors would like to express their gratitude to the referees for giving strong and very useful suggestions for improving the article.
Availability of data and materials
Not applicable.
Funding
This work is supported by the Science Research Projection of the Education Department of Fujian Province, No. JT180122, the Natural Science Foundation of Fujian Province, No. 2015J01019, and NSFC(Nos. 11021101, 11290142, 91130003, 11771149 and 11701086).
Authors’ contributions
All authors participated in drafting and checking the manuscript, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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