The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7^{th} order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.
Published in | Engineering Mathematics (Volume 7, Issue 1) |
DOI | 10.11648/j.engmath.20230701.12 |
Page(s) | 19-28 |
Creative Commons |
This is an Open Access article, 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 or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Diagonally Implicit Scheme, Seventh Order, Symplectic Runge-Kutta Method
Order | RK | SRK |
---|---|---|
1 | 1 | 1 |
2 | 2 | 1 |
3 | 4 | 3 |
4 | 8 | 4 |
5 | 17 | 6 |
6 | 37 | 10 |
7 | 85 | 21 |
T | $\mathit{r}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{\sigma}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{\gamma}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | $\mathit{\alpha}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ | Order condition |
---|---|---|---|---|---|
${\tau}_{1}$ | 1 | 1 | 1 | 1 | $\sum _{i=1}^{s}{b}_{i}=1$ |
${\tau}_{2}$ | 3 | 2 | 3 | 1 | $\sum _{i=1}^{s}{b}_{i}{c}_{i}^{2}=\frac{1}{3}$ |
${\tau}_{3}$ | 4 | 6 | 4 | 1 | $\sum _{i=1}^{s}{b}_{i}{c}_{i}^{3}=\frac{1}{4}$ |
${\tau}_{4}$ | 5 | 24 | 5 | 1 | $\sum _{i=1}^{s}{b}_{i}{c}_{i}^{4}=\frac{1}{5}$ |
${\tau}_{5}$ | 5 | 2 | 10 | 6 | $\sum _{i,j=1}^{s}{b}_{i}{c}_{i}^{2}{a}_{\mathit{ij}}{c}_{j}=\frac{1}{10}$ |
${\tau}_{6}$ | 5 | 1 | 20 | 6 | $\sum _{i,j,k=1}^{s}{b}_{i}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{c}_{k}=\frac{1}{20}$ |
${\tau}_{7}$ | 6 | 120 | 6 | 1 | $\sum _{i=1}^{s}{b}_{i}{c}_{i}^{5}=\frac{1}{6}$ |
${\tau}_{8}$ | 6 | 6 | 12 | 10 | $\sum _{i,j=1}^{s}{b}_{i}{c}_{i}^{3}{a}_{\mathit{ij}}{c}_{j}=\frac{1}{12}$ |
${\tau}_{9}$ | 6 | 2 | 24 | 15 | $\sum _{i,j,k=1}^{s}{b}_{i}{c}_{i}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{c}_{k}=\frac{1}{24}$ |
${\tau}_{10}$ | 6 | 2 | 36 | 1 | $\sum _{i,j,k=1}^{s}{b}_{i}{c}_{i}^{2}{a}_{\mathit{ij}}{{a}_{\mathit{jk}}c}_{k}=\frac{1}{36}$ |
${\tau}_{11}$ | 7 | 2 | 252 | 10 | $\sum _{i,j,k,l,m=1}^{s}{b}_{i}{a}_{\mathit{ij}}{{a}_{\mathit{jk}}c}_{k}{a}_{\mathit{il}}{a}_{\mathit{lm}}{c}_{m}=\frac{1}{252}$ |
${\tau}_{12}$ | 7 | 1 | 84 | 60 | $\sum _{i,j,k,l=1}^{s}{b}_{i}{c}_{i}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{a}_{\mathit{kl}}{c}_{l}=\frac{1}{84}$ |
${\tau}_{13}$ | 7 | 6 | 42 | 20 | $\sum _{i,j,k=1}^{s}{b}_{i}{c}_{i}^{3}{a}_{\mathit{ij}}{a}_{\mathit{jk}}{c}_{k}=\frac{1}{42}$ |
${\tau}_{14}$ | 7 | 8 | 63 | 10 | $\sum _{i,j,k=1}^{s}{b}_{i}{a}_{\mathit{ij}}{c}_{j}^{2}{a}_{\mathit{ik}}{c}_{k}^{2}=\frac{1}{63}$ |
${\tau}_{15}$ | 7 | 24 | 14 | 15 | $\sum _{i,j=1}^{s}{b}_{i}{c}_{i}^{4}{a}_{\mathit{ij}}{c}_{j}=\frac{1}{14}$ |
${\tau}_{16}$ | 7 | 4 | 28 | 45 | $\sum _{i,j,k=1}^{s}{b}_{i}{c}_{i}^{2}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{c}_{k}=\frac{1}{28}$ |
${\tau}_{17}$ | 7 | 6 | 56 | 15 | $\sum _{i,j,k,l=1}^{s}{b}_{i}{a}_{\mathit{ij}}{c}_{j}{{a}_{\mathit{ik}}c}_{k}{a}_{\mathit{il}}{c}_{l}=\frac{1}{56}$ |
${\tau}_{18}$ | 7 | 2 | 42 | 60 | $\sum _{i,j,k=1}^{s}{b}_{i}{c}_{i}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{c}_{k}^{2}=\frac{1}{42}$ |
${\tau}_{19}$ | 7 | 720 | 7 | 1 | $\sum _{i=1}^{s}{b}_{i}{c}_{i}^{6}=\frac{1}{7}$ |
${\tau}_{20}$ | 7 | 2 | 168 | 15 | $\sum _{i,j,k,l=1}^{s}{b}_{i}{a}_{\mathit{ij}}{c}_{j}{a}_{\mathit{ik}}{a}_{\mathit{kl}}{c}_{l}^{2}=\frac{1}{168}$ |
${\tau}_{21}$ | 7 | 12 | 21 | 20 | $\sum _{i,j=1}^{s}{b}_{i}{c}_{i}^{3}{a}_{\mathit{ij}}{c}_{j}^{2}=\frac{1}{21}$ |
Parameter | Value | Parameter | Value |
---|---|---|---|
${b}_{1}$ | 0.0884823 | ${c}_{1}$ | 0.0442412 |
${b}_{2}$ | 0.0898123 | ${c}_{2}$ | 0.1333885 |
${b}_{3}$ | 0.0931355 | ${c}_{3}$ | 0.2248624 |
${b}_{4}$ | 0.0996577 | ${c}_{4}$ | 0.321259 |
${b}_{5}$ | 0.112612 | ${c}_{5}$ | 0.4273938 |
${b}_{6}$ | 0.145116 | ${c}_{6}$ | 0.5562578 |
${b}_{7}$ | 0.246823 | ${c}_{7}$ | 0.7522273 |
${b}_{8}$ | -0.333089 | ${c}_{8}$ | 0.7090943 |
${b}_{9}$ | 0.24727 | ${c}_{9}$ | 0.6661848 |
${b}_{10}$ | 0.122669 | ${c}_{10}$ | 0.8511543 |
${b}_{11}$ | 0.0875886 | ${c}_{11}$ | 0.9562831 |
h | M968 | M11,7,8 |
---|---|---|
0.2 | $4.02\bullet {10}^{-6}$ | $7.43\bullet {10}^{-9}$ |
0.1 | $5.83\bullet {10}^{-8}$ | $7.65\bullet {10}^{-11}$ |
h | M968 | M11,7,8 |
---|---|---|
0.2 | $7.32\bullet {10}^{-6}$ | $1.111\bullet {10}^{-7}$ |
0.1 | $4.71\bullet {10}^{-7}$ | $1.211\bullet {10}^{-8}$ |
0.01 | $2.37\bullet {10}^{-9}$ | $3.23\bullet {10}^{-11}$ |
RK | Runge-Kutta Method |
SRK | Symplectic Runge-Kutta Method |
DISRK | Diagonally Implicit Symplectic Runge-Kutta Method |
[1] | A. Iserles. Efficient Runge-Kutta methods for Hamiltonian equations. Bulletin of the Greek Mathematical Society, 1991, 32, 3-20, |
[2] | Ch. Tsitouras. Explicit Runge-Kutta methods for starting integration of Lane-Emden problem. Appl. Mathe. Comput, 2019, 354, 353-364, |
[3] | E. Hairer, G. Wanner. Symplectic Runge-Kutta methods with real eigenvalues. BIT Nume, Mathe, 1994, 34(2), 310-312. |
[4] | E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration: Structure Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin, Germany, 1994, 2002. |
[5] | E. Hairer, C. Lubich, G. Wanner. Geometric Numerical Integration. Structure Preserving Algorithms for Ordinary Differential Equations, 2nd ed, Springer, Berlin, 2006. |
[6] | E. Hairer, S. P. Norsett, G. Wanner, Solving Ordinary Differential Equations I, second ed., Springer, Heidelberg, 1993. |
[7] | G. Avdelas, A. Konguetsof, T. E. Simos. A generator and an optimized generator of high-order hybrid explicit methods for the numerical solution of the Schrodinger equation. Part 1. Deveolpment of the basic method, J. Math. Chem, 2001, 29, 281291. |
[8] | Geng Sun. A simple way constructing symplectic Runge-Kutta methods. J. Comput. Mathe, 2000, 18(1), 61-68. |
[9] | Geng Sun. Symmetric-Adjoint and Symplectic-Adjoint Runge-Kutta Methods and Their Applications, Numer. Math. Theor. Meth. Appl, 2022, 15(2), 304-335, |
[10] | H. van de Vyer. Fourth order symplectic integration with reduced phase error, International Journal of Modern Physics C, 2008, 19(8), 1257-1268. |
[11] | J. C. Butcher. A history of Runge-Kutta methods, Applied Numerical Mathematics, 1996, 20, 257-260. |
[12] | J. C. Simo, N. Tarnow. A new energy and momentum conserving algorithm for the nonlinear dynamics of shells, Internet. J. Numer. Methods Engrg., 1994, 37, 2527-2549. |
[13] | J. M. Franco, I. Gomez. Fourth-order symmetric DIRK methods for periodic stiff problems, Numerical Algorithms, 2003, 32(24), 317-326. |
[14] | J. M. Sanz-Serna, M. P. Calvo. Numerical Hamiltonian Problems, Champman and Hall, London, 1994. |
[15] | Kaifeng Xia, Yuhao Cong, Geng Sun. Symplectic Runge-Kutta Methods of high order based on W-transformation, Journal of Applied Analysis and Computation, 2017, 7(3), |
[16] | Kang Feng, Mengzhao Qin. Symplectic Geometric Algorithms for Hamiltonian Systems, Zhejiang Publishing United Group, Zhejiang Science and Technology Publishing House, 2009, 277-302. |
[17] | K. Feng. Collected Works of Feng Kang (II), National Defence Press, Beijing, 1995. |
[18] | K. Tselios, T. E. Simos. Runge-Kutta methods with minimal dispersion and dissipation for problems arising from computational acoustics, Journal of Computational and Applied Mathematics, 2005, 175(1), 173-181. |
[19] | L. Brusa, L. Nigro. A one-step method for direct integration of structural dynamic equations, International Journal for Numerical Methods in Engineering, 1980, 15(5), 685-699. |
[20] | M. Z. Qin, M. Q. Zhang. Symplectic Runge-Kutta algorithmz for Hamilton systems, Journal of Computational Mathematics, Supplementary Issue, 1992, 205-215. |
[21] | P. J. van der Houwen, B. P. Sommejier. Phase-lag analysis of implicit Runge-Kutta methods, SIAM J. Numer. Anal, 1989, 26(1), 214-229, |
[22] | Run-de Zhang, Le-ping Yang, Wei-wei Cai. Symplectic Runge-Kutta Method Based Numerical Solution for the Hamiltonian Model of Spacecraft Relative Motion, Springer Nature Singapore Pte Ltd, 2019, |
[23] | U. Ascher, S. Reich. On some difficulties in integrating highly oscillatory Hamiltonian systems, in Computational Molecular Dynamics, Lecture Notes in Comput. Sci. Engrg. 4, Springer, Berlin, 1999, 281-296. |
[24] | V. I. Amold. Mathematical Methods of Classical Mechanics, Springer-Berlin, 1999. |
[25] | Y. B. Suris. Canonical transformation generated by methods of Runge-Kutta type for the numerical integration of the system x´´=-∂U/∂x, Zhurnal Vychisliteli Matermatiki I Mathematics, 2005, 175(1), 173-181. |
[26] | Y. H. Cong, C. X. Jiang. Diagonally Implicit Symplectic Runge-Kutta Methods of fifth and sixth order, The Scientific World Journal, 2014, |
[27] | Y. H. Cong, C. X. Jiang. Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order, The Scientific World Journal, 2014, Article ID 147801, 7, |
[28] | Y. L. Fang, Q. H. Li. A class of explicit rational symplectic integrators, Journal of Applied Analysis and Computation, 2012, 2(2), 161-171. |
[29] | Z. Kalogiratou, Th. Monovasilis. Diagonally Implicit Symplectic Runge-Kutta Methods with Special Properties, Appli. Mathe. & Infor. Scie, Feb. 2015, |
[30] | Z. Kalogiratou, T. Monovasilis, T. E. Simos. A diagonally implicit symplectic Runge-Kutta method with minimum phase-lag, in Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 11), vol. 1389 of AIP Conference Proceedings, 2011, 1977-1979, |
[31] | Z. Kalogiratou, T. Monovasilis, T. E. Simos. Diagonally implicit symplectic Runge-Kutta method with special properties, in Proceedings of the International Conference on Numerical Analysis and Applied Mathematics (ICNAAM 12), 1479 of AIP Conference Proceedings, 2012, 1387-1390, |
APA Style
Oh, T. G., Choe, J. H., Kim, J. S. (2024). Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order. Engineering Mathematics, 7(1), 19-28. https://doi.org/10.11648/j.engmath.20230701.12
ACS Style
Oh, T. G.; Choe, J. H.; Kim, J. S. Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order. Eng. Math. 2024, 7(1), 19-28. doi: 10.11648/j.engmath.20230701.12
AMA Style
Oh TG, Choe JH, Kim JS. Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order. Eng Math. 2024;7(1):19-28. doi: 10.11648/j.engmath.20230701.12
@article{10.11648/j.engmath.20230701.12, author = {Thae Gun Oh and Ji Hyang Choe and Jin Sim Kim}, title = {Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order }, journal = {Engineering Mathematics}, volume = {7}, number = {1}, pages = {19-28}, doi = {10.11648/j.engmath.20230701.12}, url = {https://doi.org/10.11648/j.engmath.20230701.12}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20230701.12}, abstract = {The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7th order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods. }, year = {2024} }
TY - JOUR T1 - Diagonally Implicit Symplectic Runge-Kutta Methods with 7th Algebraic Order AU - Thae Gun Oh AU - Ji Hyang Choe AU - Jin Sim Kim Y1 - 2024/07/03 PY - 2024 N1 - https://doi.org/10.11648/j.engmath.20230701.12 DO - 10.11648/j.engmath.20230701.12 T2 - Engineering Mathematics JF - Engineering Mathematics JO - Engineering Mathematics SP - 19 EP - 28 PB - Science Publishing Group SN - 2640-088X UR - https://doi.org/10.11648/j.engmath.20230701.12 AB - The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. Since Hamiltonian systems have good properties such as symplecticity, numerical methods that preserve these properties have attracted the great attention. In fact, the explicit Runge-Kutta methods have used due to that schemes are very simple and its computational amounts are very small. However, the explicit schemes aren’t stable so the implicit Runge-Kutta methods have widely studied. Among those implicit schemes, symplectic numerical methods were interested. It is because it has preserved the original property of the systems. So, study of the symplectic Runge-Kutta methods have performed. The typical symplectic Runge-Kutta method is the Gauss-Legendre method, whose drawback is that it is a general implicit scheme and is too computationally expensive. Despite these drawbacks, the study of the diagonally implicit symplectic Runge-Kutta methods that preserves symplecticity has attracted much attention. The symplectic Runge-Kutta method has been studied up to sixth order in the past and efforts to obtain higher order conditions and algorithms are being intensified. In many applications such as molecular dynamics as well as in space science, such as satellite relative motion studies, this method is very effective and its application is wider. In this paper, it is presented the 7th order condition and derive the corresponding optimized method. So the diagonally implicit symplectic eleven-stages Runge-Kutta method with algebraic order 7 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods. VL - 7 IS - 1 ER -