-
Previous Article
A new class of integrable Lotka–Volterra systems
- JCD Home
- This Issue
-
Next Article
Deep learning as optimal control problems: Models and numerical methods
Algebraic structure of aromatic B-series
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway |
Aromatic B-series are a generalization of B-series. Some of the algebraic structures on B-series can be defined analogically for aromatic B-series. This paper derives combinatorial formulas for the composition and substitution laws for aromatic B-series.
References:
[1] |
G. Bogfjellmo and A. Schmeding,
The Lie group structure of the Butcher group, Found. Comput. Math., 17 (2017), 127-159.
doi: 10.1007/s10208-015-9285-5. |
[2] |
C. Brouder,
Runge–Kutta methods and renormalization, European Physical J. C-Particles and Fields, 12 (2000), 521-534.
doi: 10.1007/s100529900235. |
[3] |
J. C. Butcher,
Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963), 185-201.
doi: 10.1017/S1446788700027932. |
[4] |
J. C. Butcher,
An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106.
doi: 10.1090/S0025-5718-1972-0305608-0. |
[5] |
D. Calaque, K. Ebrahimi-Fard and D. Manchon,
Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series, Adv. in Appl. Math., 47 (2011), 282-308.
doi: 10.1016/j.aam.2009.08.003. |
[6] |
P. Chartier, E. Hairer and G. Vilmart,
Algebraic structures of B-series, Found. Comput. Math., 10 (2010), 407-427.
doi: 10.1007/s10208-010-9065-1. |
[7] |
P. Chartier and A. Murua,
Preserving first integrals and volume forms of additively split systems, IMA J. Numer. Anal., 27 (2007), 381-405.
doi: 10.1093/imanum/drl039. |
[8] |
A. Connes and D. Kreimer,
Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199 (1998), 203-242.
doi: 10.1007/s002200050499. |
[9] |
K. Feng and Z. J. Shang,
Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.
doi: 10.1007/s002110050153. |
[10] |
J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. |
[11] |
E. Hairer and G. Wanner,
On the Butcher group and general multi-value methods, Computing (Arch. Elektron. Rechnen), 13 (1974), 1-15.
doi: 10.1007/BF02268387. |
[12] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[13] |
A. Iserles, G. R. W. Quispel and P. S. P. Tse,
B-series methods cannot be volume-preserving, BIT, 47 (2007), 351-378.
doi: 10.1007/s10543-006-0114-8. |
[14] |
D. Manchon, Algebraic background for numerical methods, control theory and renormalization, preprint, arXiv: math/1501.07205. |
[15] |
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier,
Butcher series: A story of rooted trees and numerical methods for evolution equations, Asia Pac. Math. Newsl., 7 (2017), 1-11.
|
[16] |
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier,
B-series methods are exactly the affine equivariant methods, Numer. Math., 133 (2016), 599-622.
doi: 10.1007/s00211-015-0753-2. |
[17] |
H. Munthe-Kaas and O. Verdier,
Aromatic Butcher series, Found. Comput. Math., 16 (2016), 183-215.
doi: 10.1007/s10208-015-9245-0. |
[18] |
A. Murua,
Formal series and numerical integrators, part Ⅰ: Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251.
doi: 10.1016/S0168-9274(98)00064-6. |
[19] |
G. M. Poore, Reproducible documents with pythontex, in Proceedings of the 12th Python in Science Conference, 2013, 78–84. |
[20] |
G. R. W. Quispel,
Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30.
doi: 10.1016/0375-9601(95)00586-R. |
[21] |
J. M. Sanz-Serna and A. Murua, Formal series and numerical integrators: Some history and
some new techniques, in Proceedings of the 8th International Congress on Industrial and
Applied Mathematics, Higher Ed. Press, Beijing, 2015, 311-331. |
[22] |
M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. |
show all references
References:
[1] |
G. Bogfjellmo and A. Schmeding,
The Lie group structure of the Butcher group, Found. Comput. Math., 17 (2017), 127-159.
doi: 10.1007/s10208-015-9285-5. |
[2] |
C. Brouder,
Runge–Kutta methods and renormalization, European Physical J. C-Particles and Fields, 12 (2000), 521-534.
doi: 10.1007/s100529900235. |
[3] |
J. C. Butcher,
Coefficients for the study of Runge-Kutta integration processes, J. Austral. Math. Soc., 3 (1963), 185-201.
doi: 10.1017/S1446788700027932. |
[4] |
J. C. Butcher,
An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106.
doi: 10.1090/S0025-5718-1972-0305608-0. |
[5] |
D. Calaque, K. Ebrahimi-Fard and D. Manchon,
Two interacting Hopf algebras of trees: A Hopf-algebraic approach to composition and substitution of B-series, Adv. in Appl. Math., 47 (2011), 282-308.
doi: 10.1016/j.aam.2009.08.003. |
[6] |
P. Chartier, E. Hairer and G. Vilmart,
Algebraic structures of B-series, Found. Comput. Math., 10 (2010), 407-427.
doi: 10.1007/s10208-010-9065-1. |
[7] |
P. Chartier and A. Murua,
Preserving first integrals and volume forms of additively split systems, IMA J. Numer. Anal., 27 (2007), 381-405.
doi: 10.1093/imanum/drl039. |
[8] |
A. Connes and D. Kreimer,
Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199 (1998), 203-242.
doi: 10.1007/s002200050499. |
[9] |
K. Feng and Z. J. Shang,
Volume-preserving algorithms for source-free dynamical systems, Numer. Math., 71 (1995), 451-463.
doi: 10.1007/s002110050153. |
[10] |
J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. |
[11] |
E. Hairer and G. Wanner,
On the Butcher group and general multi-value methods, Computing (Arch. Elektron. Rechnen), 13 (1974), 1-15.
doi: 10.1007/BF02268387. |
[12] |
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006.
doi: 10.1007/3-540-30666-8. |
[13] |
A. Iserles, G. R. W. Quispel and P. S. P. Tse,
B-series methods cannot be volume-preserving, BIT, 47 (2007), 351-378.
doi: 10.1007/s10543-006-0114-8. |
[14] |
D. Manchon, Algebraic background for numerical methods, control theory and renormalization, preprint, arXiv: math/1501.07205. |
[15] |
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier,
Butcher series: A story of rooted trees and numerical methods for evolution equations, Asia Pac. Math. Newsl., 7 (2017), 1-11.
|
[16] |
R. I. McLachlan, K. Modin, H. Munthe-Kaas and O. Verdier,
B-series methods are exactly the affine equivariant methods, Numer. Math., 133 (2016), 599-622.
doi: 10.1007/s00211-015-0753-2. |
[17] |
H. Munthe-Kaas and O. Verdier,
Aromatic Butcher series, Found. Comput. Math., 16 (2016), 183-215.
doi: 10.1007/s10208-015-9245-0. |
[18] |
A. Murua,
Formal series and numerical integrators, part Ⅰ: Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251.
doi: 10.1016/S0168-9274(98)00064-6. |
[19] |
G. M. Poore, Reproducible documents with pythontex, in Proceedings of the 12th Python in Science Conference, 2013, 78–84. |
[20] |
G. R. W. Quispel,
Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30.
doi: 10.1016/0375-9601(95)00586-R. |
[21] |
J. M. Sanz-Serna and A. Murua, Formal series and numerical integrators: Some history and
some new techniques, in Proceedings of the 8th International Congress on Industrial and
Applied Mathematics, Higher Ed. Press, Beijing, 2015, 311-331. |
[22] |
M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. |
![]() |
![]() |
[1] |
Karl Petersen, Ibrahim Salama. Entropy on regular trees. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4453-4477. doi: 10.3934/dcds.2020186 |
[2] |
Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237 |
[3] |
Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems and Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1 |
[4] |
Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 623-641. doi: 10.3934/dcds.2021131 |
[5] |
Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks and Heterogeneous Media, 2021, 16 (2) : 317-339. doi: 10.3934/nhm.2021008 |
[6] |
Bertrand Maury, Delphine Salort, Christine Vannier. Trace theorems for trees and application to the human lungs. Networks and Heterogeneous Media, 2009, 4 (3) : 469-500. doi: 10.3934/nhm.2009.4.469 |
[7] |
Xavier Dubois de La Sablonière, Benjamin Mauroy, Yannick Privat. Shape minimization of the dissipated energy in dyadic trees. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 767-799. doi: 10.3934/dcdsb.2011.16.767 |
[8] |
Thierry Coulbois. Fractal trees for irreducible automorphisms of free groups. Journal of Modern Dynamics, 2010, 4 (2) : 359-391. doi: 10.3934/jmd.2010.4.359 |
[9] |
Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial and Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108 |
[10] |
Chris Bernhardt. Vertex maps for trees: Algebra and periods of periodic orbits. Discrete and Continuous Dynamical Systems, 2006, 14 (3) : 399-408. doi: 10.3934/dcds.2006.14.399 |
[11] |
Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2823-2835. doi: 10.3934/dcdss.2020464 |
[12] |
Erick Limas. An application of minimal spanning trees and hierarchical trees to the study of Latin American exchange rates. Journal of Dynamics and Games, 2019, 6 (2) : 131-148. doi: 10.3934/jdg.2019010 |
[13] |
Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419 |
[14] |
Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581 |
[15] |
JÓzsef Balogh, Hoi Nguyen. A general law of large permanent. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5285-5297. doi: 10.3934/dcds.2017229 |
[16] |
Sergei Avdonin, Pavel Kurasov, Marlena Nowaczyk. Inverse problems for quantum trees II: Recovering matching conditions for star graphs. Inverse Problems and Imaging, 2010, 4 (4) : 579-598. doi: 10.3934/ipi.2010.4.579 |
[17] |
Patrick Joly, Maryna Kachanovska, Adrien Semin. Wave propagation in fractal trees. Mathematical and numerical issues. Networks and Heterogeneous Media, 2019, 14 (2) : 205-264. doi: 10.3934/nhm.2019010 |
[18] |
Ambroise Vest. On the structural properties of an efficient feedback law. Evolution Equations and Control Theory, 2013, 2 (3) : 543-556. doi: 10.3934/eect.2013.2.543 |
[19] |
Alberto Bressan, Graziano Guerra. Shift-differentiabilitiy of the flow generated by a conservation law. Discrete and Continuous Dynamical Systems, 1997, 3 (1) : 35-58. doi: 10.3934/dcds.1997.3.35 |
[20] |
Marilena Filippucci, Andrea Tallarico, Michele Dragoni. Simulation of lava flows with power-law rheology. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 677-685. doi: 10.3934/dcdss.2013.6.677 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]