American Institute of Mathematical Sciences

Advanced
December  2019, 6(2): 199-222. doi: 10.3934/jcd.2019010

Algebraic structure of aromatic B-series

Geir Bogfjellmo

Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

Received  April 2019 Revised  September 2019 Published  November 2019

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.

Keywords: Aromatic series, aromatic trees, composition law, substitution law, coalgebra of aromatic trees, coproduct.
Mathematics Subject Classification: 37C10, 41A58, 16T05.
Citation: Geir Bogfjellmo. Algebraic structure of aromatic B-series. Journal of Computational Dynamics, 2019, 6 (2) : 199-222. doi: 10.3934/jcd.2019010
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.  Google Scholar

[2]

C. Brouder, Runge–Kutta methods and renormalization, European Physical J. C-Particles and Fields, 12 (2000), 521-534.  doi: 10.1007/s100529900235.  Google Scholar

[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.  Google Scholar

[4]

J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106.  doi: 10.1090/S0025-5718-1972-0305608-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199 (1998), 203-242.  doi: 10.1007/s002200050499.  Google Scholar

[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.  Google Scholar

[10]

J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Manchon, Algebraic background for numerical methods, control theory and renormalization, preprint, arXiv: math/1501.07205. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

H. Munthe-Kaas and O. Verdier, Aromatic Butcher series, Found. Comput. Math., 16 (2016), 183-215.  doi: 10.1007/s10208-015-9245-0.  Google Scholar

[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.  Google Scholar

[19]

G. M. Poore, Reproducible documents with pythontex, in Proceedings of the 12th Python in Science Conference, 2013, 78–84. Google Scholar

[20]

G. R. W. Quispel, Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30.  doi: 10.1016/0375-9601(95)00586-R.  Google Scholar

[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.  Google Scholar

[22]

M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.  Google Scholar

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.  Google Scholar

[2]

C. Brouder, Runge–Kutta methods and renormalization, European Physical J. C-Particles and Fields, 12 (2000), 521-534.  doi: 10.1007/s100529900235.  Google Scholar

[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.  Google Scholar

[4]

J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79-106.  doi: 10.1090/S0025-5718-1972-0305608-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., 199 (1998), 203-242.  doi: 10.1007/s002200050499.  Google Scholar

[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.  Google Scholar

[10]

J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. Manchon, Algebraic background for numerical methods, control theory and renormalization, preprint, arXiv: math/1501.07205. Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[17]

H. Munthe-Kaas and O. Verdier, Aromatic Butcher series, Found. Comput. Math., 16 (2016), 183-215.  doi: 10.1007/s10208-015-9245-0.  Google Scholar

[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.  Google Scholar

[19]

G. M. Poore, Reproducible documents with pythontex, in Proceedings of the 12th Python in Science Conference, 2013, 78–84. Google Scholar

[20]

G. R. W. Quispel, Volume-preserving integrators, Phys. Lett. A, 206 (1995), 26-30.  doi: 10.1016/0375-9601(95)00586-R.  Google Scholar

[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.  Google Scholar

[22]

M. E. Sweedler, Hopf Algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969.  Google Scholar

Table 1.  The coproduct $\Delta_{\mathcal{AT}}$
Table Options

Download as excel

Table 2.  Substitution law (1 root)
Table Options

Download as excel

Table 3.  Substitution law (0 roots)
Table Options

Download as excel

[1]

Karl Petersen, Ibrahim Salama. Entropy on regular trees. Discrete & 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 & 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 & 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 & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021131
[5]

Sergei Avdonin, Julian Edward. An inverse problem for quantum trees with observations at interior vertices. Networks & 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 & 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 & 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 & 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 & 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 & 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 & Games, 2019, 6 (2) : 131-148. doi: 10.3934/jdg.2019010
[13]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Continuous Dynamical Systems - S, 2013, 6 (3) : 677-685. doi: 10.3934/dcdss.2013.6.677

 Impact Factor: 

Tools

Metrics

  • PDF downloads (200)
  • HTML views (334)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy