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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

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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 $\Delta_{\mathcal{AT}}$
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Table 2.  Substitution law (1 root)
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Table 3.  Substitution law (0 roots)
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