A new approach to estimating a numerical solution in the error embedded correction framework
- Philsu Kim^{1},
- Xiangfan Piao^{2},
- WonKyu Jung^{3} and
- Sunyoung Bu^{4}Email author
https://doi.org/10.1186/s13662-018-1619-6
© The Author(s) 2018
Received: 18 September 2017
Accepted: 26 April 2018
Published: 8 May 2018
Abstract
On the basis of the error correction method developed recently, an algorithm, so-called error embedded error correction method, is proposed for initial value problems. Two deferred equations are used to approximate the solution and the error, respectively, at each integration step. For the solution, the deferred equation, which is based on a modified Euler’s polygon including the information of both the solution and its estimated error at the previous integration step, is solved with the classical fourth-order Runge–Kutta method. For the error, the deferred equation, which is based on a local Hermite cubic polynomial with three pieces of information—the solution, its estimated error at the previous step, and the constructed solution—is solved by the seventh-order Runge–Kutta–Fehlberg method. The constructed algorithm controls the error and possesses a good behavior of error bound in a long time simulation. Numerical experiments are presented to validate the proposed algorithm.
Keywords
1 Introduction
In the existing schemes, the estimated error \(e_{m}\) obtained from the previous time step \([t_{m-1}, t_{m}]\) is mainly used only for choosing an appropriate next time step size in most algorithms. Also, the solution \(\phi_{m+1}\) at time \(t_{m+1}\) is calculated with an initial value which is a solution \(\phi_{m}\) at the previous time \(t_{m}\). That is, \(\phi_{m}\) is assumed to be the exact initial condition for \(\phi_{m+1}\) despite the existence of the local truncation error \(e_{m}\), which leads to accumulation of the error as the time is increasing. In order to control the accumulation error, smaller integration steps or special step size controllers are sometimes required, especially for a long time simulation or stiff systems. Nevertheless, the most existing methods cannot fully resolve the error control to get a given tolerance so that it is difficult to get reliable results at stringent tolerances (for example, see [12, 13]).
This paper is organized as follows. In Sect. 2, we describe the methodology to formulate and control the solution and error formulas based on ECM. In Sect. 3, we give a concrete analysis of the convergence for the developed EEECM. Several numerical results are presented in Sect. 4 to give both the numerical evidences for the theoretical analysis and the numerical effectiveness of EEECM. Finally, in Sect. 5, a summary for EEECM and some discussion for further works are given.
2 Derivation of algorithm
Lemma 1
Proof
Remark 1
Remark that 16 evaluations of the Hermite interpolation and its derivatives are required for algorithm (22). However, by introducing Lemma 1, only one evaluation of the Hermite interpolation is required. It is remarkable.
Remark 2
Remark 3
(Geometric interpretation)
3 Convergence analysis
Lemma 2
Proof
From equation (39) together with (41) in the above lemma, we have the following corollary.
Corollary 1
Proof
By directly substituting (41) into (39) and expanding the resulted equation in ascending order of h with the aid of the identity \(({\mathcal {A}}\mathbf{c}^{3})_{i} = \frac{\mathbf{c}_{i}^{4}}{4}, i\geq6 \), one may get the required equation (46). □
Substituting expansion (46) into the sum of F defined by (38) leads to the following theorem.
Theorem 1
Proof
Theorem 2
Assume that the present method (35) satisfies the Lipschitz condition (54) and the slope function f is sufficiently smooth. Then, for the IVP (1), algorithm (35) has the rate of convergence \(\mathcal {O} (h^{7} )\).
Remark 4
Theorem 2 shows that the estimated error \(e_{m+1}\) in algorithm (33) exactly estimates the coefficients of Taylor’s expansion about h of the error \(E_{m+1}:=\phi(t_{m+1})-\phi_{m+1}\) up to the 7th order term, whereas the embedded RKF78 exactly estimates the 8th order term only. Also, unlike the existing embedded schemes, the estimated error \(e_{m}\) is embedded in the algorithm EEECM itself by considering as an initial value at each time interval. It turns out that the proposed algorithm (33) is more efficient in a long time simulation, which is shown throughout several numerical results (see Sect. 4).
4 Numerical results
4.1 Simple problems
In this subsection, we will show the efficiency of EEECM with two simple IVPs. One is a well-known simple harmonic oscillator. The other is knowing that the global error control is quite difficult [12]. Details of each problem will be explained in each subsection.
Example 1
Convergence order of EEECM for solving a simple harmonic oscillator
Step-size | Error | Rate |
---|---|---|
0.5 | 2.7007e–006 | |
0.25 | 1.8878e–008 | 7.160437 |
0.125 | 1.3484e–010 | 7.129298 |
0.0625 | 9.9618e–013 | 7.080680 |
0.03125 | 7.0429e–015 | 7.144086 |
Example 2
As shown in the first example, we also calculate the time cost required to obtain the desired accuracy by varying tolerances from \(\mathit{tol}=1\text{e--}5\) to \(\mathit{tol}=1\text{e--}10\) and plot the numerical results in Fig. 4(b). In this example, the numerical results show that the proposed scheme obtains the most accurate solution for each fixed CPU time. In particular, one can see that the proposed method achieves the required accuracies within the given tolerances, whereas all existing methods fail to achieve this requirement. Even the absolute errors of the other methods at the final time achieve about only half order for the desired accuracy even though the required CPU time is small compared to our method. That is, one may claim that our method well controls the global error within the given tolerances for this complicated system.
4.2 Hamiltonian system
Formally, a Hamiltonian system is a dynamical system completely described by the scalar function H, the Hamiltonian. Firstly, we solve a simple pendulum problem to show how well EEECM can conserve the total energy H. Secondly, we test a two-body Kepler problem to confirm that the proposed method is well fit for the Hamiltonian system.
4.2.1 Pendulum problem
4.2.2 Kepler problem
5 Conclusion and further discussion
A new error control strategy for non-stiff problems is developed within the ECM framework. Unlike the traditional way to approximate solutions in an explicit single step method, we suggest a methodology that contains the estimated error at each integration step and enables us to control the bound of the local truncation error for a long time simulation. Throughout several numerical results, it is shown that the proposed method obtains a uniform-like error bound, which is outstanding compared with existing numerical methods. Also, it is seen that like symplectic methods, the proposed scheme preserves the invariants such as the energy and angular momentum in Hamiltonian systems.
In order to fully explore the efficiency of EEECM, several extended issues are currently being pursued. One of them is to optimize the number of function evaluations to reduce the computational cost such as the existing embedded algorithms. Another issue is to investigate strategies for selecting time integration step size, since an adaptive time stepping is necessary to find efficient solutions for a long time simulation. The proposed method is developed only for non-stiff problems, and we solved simple Hamiltonian systems. Hence, the other challenge is to extend the idea of the proposed method into stiff systems. Additionally, the generalization of the proposed idea will be applied to many physical problems expressed by partial differential equations (PDEs). Results along these directions will be reported in the future.
Declarations
Acknowledgements
The authors would like to express their gratitude to the reviewers and the editor for valuable suggestions and comments.
Funding
The first author Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number 2016R1A2B2011326). Also, the corresponding author Bu was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number 2016R1D1A1B03930734). The second author Piao was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (grant number 2017R1C1B1002370).
Authors’ contributions
PK and XP provided the basic idea of this work and developed all theory needed in this manuscript. WJ simulated the numerical examples, and the corresponding author SB completed the proofs for all the theorems in this manuscript and wrote the manuscripts. All authors 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
References
- Hairer, E., Norsett, S.P., Wanner, G.: Solving Ordinary Differential Equations. I Nonstiff. Springer Series in Computational Mathematics. Springer, Berlin (1993) MATHGoogle Scholar
- Hairer, E., Wanner, G.: Solving Ordinary Differential Equations. II Stiff and Differential-Algebraic Problems. Springer Series in Computational Mathematics. Springer, Berlin (1996) MATHGoogle Scholar
- Calvo, M.P., Hairer, E.: Accurate long-term integration of dynamical systems. Appl. Numer. Math. 18, 95–105 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Hairer, E.: Long-time integration of non-stiff and oscillatory Hamiltonian systems. AIP Conf. Proc. 1168(1), 3–6 (2009) View ArticleMATHGoogle Scholar
- Tiwari, S., Kumar, M.: An initial value technique to solve two-point non-linear singularly perturbed boundary value problems. Appl. Comput. Math. 14(2), 150–157 (2015) MathSciNetMATHGoogle Scholar
- Dormand, J.R., Prince, P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6(1), 19–26 (1980) MathSciNetView ArticleMATHGoogle Scholar
- Estep, D.: A posteriori error bounds and global error control for approximation of ordinary differential equations. SIAM J. Numer. Anal. 32(1), 1–48 (1995) MathSciNetView ArticleMATHGoogle Scholar
- Gustafsson, K.: Control-theoretic techniques for stepsize selection in implicit Runge–Kutta methods. ACM Trans. Math. Softw. 20(4), 496–517 (1994) MathSciNetView ArticleMATHGoogle Scholar
- Johnson, C.: Error estimates and adaptive time-step control for a class of one-step methods for stiff ordinary differential equations. SIAM J. Numer. Anal. 25(4), 908–926 (1988) MathSciNetView ArticleMATHGoogle Scholar
- Kavetski, D., Binning, P., Sloan, S.W.: Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards equation. Adv. Water Resour. 24, 595–605 (2001) View ArticleGoogle Scholar
- Kulikov, G.Y.: Global error control in adaptive nordsieck methods. SIAM J. Sci. Comput. 34(2), 839–860 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Kulikov, G.Y., Weiner, R.: Global error estimation and control in linearly-implicit parallel two-step peer W-methods. Comput. Appl. Math. 236, 1226–1239 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Shampine, L.F.: Error estimation and control for ODEs. J. Sci. Comput. 25(1), 3–16 (2005) MathSciNetView ArticleMATHGoogle Scholar
- Pereyra, V.: Iterated deferred correction for nonlinear boundary value problems. Numer. Math. 11, 111–125 (1968) MathSciNetView ArticleMATHGoogle Scholar
- Zadunaisky, P.E.: On the estimation of errors propagated in the numerical integration of ordinary differential equations. Numer. Math. 27, 21–40 (1976) MathSciNetView ArticleMATHGoogle Scholar
- Bu, S., Huang, J., Minion, M.L.: Semi-implicit Krylov deferred correction methods for differential algebraic equations. Math. Comput. 81(280), 2127–2157 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Bu, S., Lee, J.: An enhanced parareal algorithm based on the deferred correction methods for a stiff system. J. Comput. Appl. Math. 255(1), 297–305 (2014) MathSciNetView ArticleMATHGoogle Scholar
- Dutt, A., Greengard, L., Rokhlin, V.: Spectral deferred correction methods for ordinary differential equations. BIT Numer. Math. 40(2), 241–266 (2000) MathSciNetView ArticleMATHGoogle Scholar
- Huang, J., Jia, J., Minion, M.L.: Accelerating the convergence of spectral deferred correction methods. J. Comput. Phys. 214(2), 633–656 (2006) MathSciNetView ArticleMATHGoogle Scholar
- Huang, J., Jia, J., Minion, M.L.: Arbitrary order Krylov deferred correction methods for differential algebraic equations. J. Comput. Phys. 221(2), 739–760 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Kim, P., Piao, X., Kim, S.D.: An error corrected Euler method for solving stiff problems based on Chebyshev collocation. SIAM J. Numer. Anal. 49, 2211–2230 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Kim, S.D., Piao, X., Kim, D.H., Kim, P.: Convergence on error correction methods for solving initial value problems. J. Comput. Appl. Math. 236(17), 4448–4461 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Kim, S.D., Kim, P.: Exponentially fitted error correction methods for solving initial value problems. Kyungpook Math. J. 52, 167–177 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Kim, P., Lee, E., Kim, S.D.: Simple ECEM algorithms using function values only. Kyungpook Math. J. 53, 573–591 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Atkinson, K.E.: An Introduction to Numerical Analysis. Wiley, New York (1989) MATHGoogle Scholar
- Fehlberg, E.: Classical fifth-, sixth-, seventh-, and eighth-order Runge–Kutta formulas with stepsize control. In: NASA; for Sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield (1968) Google Scholar
- Shampine, L.F.: Vectorized solution of ODEs in MATLAB. Scalable Comput.: Pract. Experience 10, 337–345 (2010) Google Scholar
- Gear, C.W.: Numerical Initial Value Problems in Ordinary Differential Equations. Prentice Hall, New York (1971) MATHGoogle Scholar
- Brugnano, L., Iavernaro, F., Trigiante, D.: Energy- and quadratic invariants-preserving integrators based upon Gauss collocation formulae. SIAM J. Numer. Anal. 50, 2897–2916 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differenital Equations, 2nd edn. Springer Series in Computational Mathematics. Springer, Berlin (2006) MATHGoogle Scholar