Search

Your search keyword '"formal series"' showing total 8 results
Start Over Descriptor "formal series" Topic averaging
8 results on '"formal series"'

1. Averaging of highly-oscillatory transport equations.

Author

Chartier, Philippe, Crouseilles, Nicolas, Lemou, Mohammed, and Méhats, Florian

Subjects
TRANSPORT equation, ORDINARY differential equations, VLASOV equation, MAGNETIC fields, ELECTRIC fields, NORMAL forms (Mathematics)
Abstract

In this paper, we develop a new strategy aimed at obtaining high-order asymptotic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature are re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models. [ABSTRACT FROM AUTHOR]

Published
2020
Full Text
View/download PDF

2. Averaging of highly-oscillatory transport equations

Author

Nicolas Crouseilles, Florian Méhats, Philippe Chartier, Mohammed Lemou, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Multi-scale numerical geometric schemes (MINGUS), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre National de la Recherche Scientifique (CNRS), ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques(2014), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

Subjects
82B40, 01 natural sciences, 010305 fluids & plasmas, Vlasov equation, normal form, formal series, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], 0103 physical sciences, strong magnetic field, 35Q83, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics, averaging, Numerical Analysis, 010102 general mathematics, Mathematical analysis, regime, State (functional analysis), highly-oscillatory, Magnetic field, Change of variables (PDE), Exact solutions in general relativity, transport equation, Modeling and Simulation, Ordinary differential equation, Mathematics Subject Classification (2010): 34C29, Convection–diffusion equation, Constant (mathematics)
Abstract

International audience; In this paper, we develop a new strategy aimed at obtaining high-order asymp-totic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature and re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

Published
2020
Full Text
View/download PDF

3. Higher-Order Averaging, Formal Series and Numerical Integration III: Error Bounds.

Author

Chartier, P., Murua, A., and Sanz-Serna, J.

Subjects
*NUMERICAL integration, *INTEGRATORS, *HAMILTONIAN systems, *INTEGRALS, *INTEGRAL calculus
Abstract

In earlier papers, it has been shown how formal series like those used nowadays to investigate the properties of numerical integrators may be used to construct high-order averaged systems or formal first integrals of Hamiltonian problems. With the new approach the averaged system (or the formal first integral) may be written down immediately in terms of (i) suitable basis functions and (ii) scalar coefficients that are computed via simple recursions. Here we show how the coefficients/basis functions approach may be used advantageously to derive exponentially small error bounds for averaged systems and approximate first integrals. [ABSTRACT FROM AUTHOR]

Published
2015
Full Text
View/download PDF

4. Higher-Order Averaging, Formal Series and Numerical Integration II: The Quasi-Periodic Case.

Author

Chartier, P., Murua, A., and Sanz-Serna, J.

Subjects
*OSCILLATION theory of differential equations, *NUMERICAL integration, *NUMERICAL analysis, *AVERAGING principle, *MATHEMATICS
Abstract

The paper considers non-autonomous oscillatory systems of ordinary differential equations with d≥1 non-resonant constant frequencies ω,..., ω. Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all given ω,..., ω. The new method may be applied to obtain a variety of averaged systems. In particular, we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure, because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (respectively, of a divergence-free) system results in a canonical (respectively, in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies. [ABSTRACT FROM AUTHOR]

Published
2012
Full Text
View/download PDF

5. Higher-Order Averaging, Formal Series and Numerical Integration I: B-series.

Author

Chartier, P., Murua, A., and Sanz-Serna, J. M.

Subjects
*ARITHMETIC mean, *LARYNGOSTROBOSCOPY, *HAMILTONIAN systems, *INTEGRATORS, *EQUATIONS
Abstract

We show how B-series may be used to derive in a systematic way the analytical expressions of the high-order stroboscopic averaged equations that approximate the slow dynamics of highly oscillatory systems. For first-order systems we give explicitly the form of the averaged systems with $\mathcal{O}(\epsilon^{j})$ errors, j=1,2,3 (2 π ε denotes the period of the fast oscillations). For second-order systems with large $\mathcal{O}(\epsilon^{-1})$ forces, we give the explicit form of the averaged systems with $\mathcal{O}(\epsilon^{j})$ errors, j=1,2. A variant of the Fermi-Pasta-Ulam model and the inverted Kapitsa pendulum are used as illustrations. For the former it is shown that our approach establishes the adiabatic invariance of the oscillatory energy. Finally we use B-series to analyze multiscale numerical integrators that implement the method of averaging. We construct integrators that are able to approximate not only the simplest, lowest-order averaged equation but also its high-order counterparts. [ABSTRACT FROM AUTHOR]

Published
2010
Full Text
View/download PDF

6. Word Series for Dynamical Systems and Their Numerical Integrators

Author

Jesús María Sanz-Serna, Ander Murua, and Ministerio de Economía y Competitividad (España)

Subjects
Binary splitting, Power series, Matemáticas, Function series, 010103 numerical & computational mathematics, Normal forms, 01 natural sciences, Series acceleration, Trees, Averaging, B-series, Formal series, 0101 mathematics, Mathematics, Formal power series, Series (mathematics), Applied Mathematics, Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing), Neumann series, Splitting algorithms, 010101 applied mathematics, Algebra, Hamiltonian problems, Computational Mathematics, Computational Theory and Mathematics, Hopf algebras, Oscillatory differential-equations, Word series, Computer Science::Formal Languages and Automata Theory, Analysis, Word (computer architecture)
Abstract

We study word series and extended word series, classes of formal series for the analysis of some dynamical systems and their discretizations. These series are similar to but more compact than B-series. They may be composed among themselves by means of a simple rule. While word series have appeared before in the literature, extended word series are introduced in this paper. We exemplify the use of extended word series by studying the reduction to normal form and averaging of some perturbed integrable problems. We also provide a detailed analysis of the behavior of splitting numerical methods for those problems., Ministerio de Economía, Industria y Competitividad (Projects MTM2013-46553-C3-2-P and MTM2013-46553-C3-1-P)

Published
2017
Full Text
View/download PDF

7. Averaging of highly-oscillatory transport equations

Author

Chartier , Philippe, Crouseilles , Nicolas, Lemou , Mohammed, Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Invariant Preserving SOlvers ( IPSO ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Centre National de la Recherche Scientifique ( CNRS ), and ANR-14-CE23-0007,MOONRISE,MOdèles, Oscillations et SchEmas NUmeriques ( 2014 )

Subjects
averaging, 82B40, [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], regime, Numerical Analysis (math.NA), [ PHYS.PHYS.PHYS-PLASM-PH ] Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], 34C29, 82B40, 35Q83, highly-oscillatory, Vlasov equation, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], formal series, normal form, transport equation, FOS: Mathematics, strong magnetic field, 35Q83, Mathematics - Numerical Analysis, Mathematics Subject Classification (2010): 34C29
Abstract

In this paper, we develop a new strategy aimed at obtaining high-order asymp-totic models for transport equations with highly-oscillatory solutions. The technique relies upon recent developments averaging theory for ordinary differential equations, in particular normal form expansions in the vanishing parameter. Noteworthy, the result we state here also allows for the complete recovery of the exact solution from the asymptotic model. This is done by solving a companion transport equation that stems naturally from the change of variables underlying high-order averaging. Eventually, we apply our technique to the Vlasov equation with external electric and magnetic fields. Both constant and non-constant magnetic fields are envisaged, and asymptotic models already documented in the literature and re-derived using our methodology. In addition, it is shown how to obtain new high-order asymptotic models.

Published
2016

8. Higher-order averaging, formal series and numerical integration II: the quasi-periodic case

Author

Ander Murua, Philippe Chartier, Jesús María Sanz-Serna, Invariant Preserving SOlvers ( IPSO ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ) -Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique ( Inria ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Konputazio Zientziak eta A.A. Saila [Spain], Universidad del Pais Vasco / Euskal Herriko Unibertsitatea ( UPV/EHU ), Departamento de Matemática Aplicada [Valladolid] ( MA ), Universidad de Valladolid [Valladolid] ( UVa ), Invariant Preserving SOlvers (IPSO), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Universidad del Pais Vasco / Euskal Herriko Unibertsitatea [Espagne] (UPV/EHU), Departamento de Matemática Aplicada [Valladolid] (MA), Universidad de Valladolid [Valladolid] (UVa), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
high-order averaging, quasi-stroboscopic averaging, highly oscillatory problems, Scalar (mathematics), near-integrable systems, 010103 numerical & computational mathematics, 01 natural sciences, Averaging, B-series, formal series, Lie algebras, Lie algebra, 0101 mathematics, Harmonic oscillator, Mathematics, Lie groups, Applied Mathematics, Numerical analysis, first-integrals, Mathematical analysis, Lie group, trees, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Numerical integration, 010101 applied mathematics, Computational Mathematics, Hamiltonian problems, Computational Theory and Mathematics, Ordinary differential equation, Integrator, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The paper considers non-autonomous oscillatory systems of ordinary differential equations with d≥1 non-resonant constant frequencies ω 1,…,ω d . Formal series like those used nowadays to analyze the properties of numerical integrators are employed to construct higher-order averaged systems and the required changes of variables. With the new approach, the averaged system and the change of variables consist of vector-valued functions that may be written down immediately and scalar coefficients that are universal in the sense that they do not depend on the specific system being averaged and may therefore be computed once and for all given ω 1,…,ω d . The new method may be applied to obtain a variety of averaged systems. In particular, we study the quasi-stroboscopic averaged system characterized by the property that the true oscillatory solution and the averaged solution coincide at the initial time. We show that quasi-stroboscopic averaging is a geometric procedure, because it is independent of the particular choice of co-ordinates used to write the given system. As a consequence, quasi-stroboscopic averaging of a canonical Hamiltonian (respectively, of a divergence-free) system results in a canonical (respectively, in a divergence-free) averaged system. We also study the averaging of a family of near-integrable systems where our approach may be used to construct explicitly d formal first integrals for both the given system and its quasi-stroboscopic averaged version. As an application we construct three first integrals of a system that arises as a nonlinear perturbation of five coupled harmonic oscillators with one slow frequency and four resonant fast frequencies.

Published
2012
Full Text
View/download PDF

Catalog

Books, media, physical & digital resources
See catalog results