Table 1.
The coproduct $\Delta_{\mathcal{AT}}$
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December
2019, 6(2): 199-222.
doi: 10.3934/jcd.2019010
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.
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. |
| [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. |
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