American Institute of Mathematical Sciences

Advanced
December  2020, 13(6): 1107-1133. doi: 10.3934/krm.2020039

Averaging of highly-oscillatory transport equations

Philippe Chartier 1,, , Nicolas Crouseilles 1, , Mohammed Lemou 2, and  Florian Méhats 3,

1. 

Univ Rennes, Inria, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France
2. 

Univ Rennes, CNRS, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France
3. 

Univ Rennes, Irmar, Mingus Team, Campus de Beaulieu, F-35042 Rennes, France

* Corresponding author: Philippe Chartier

Received  February 2020 Revised  June 2020 Published  September 2020

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.

Keywords: Averaging, formal series, normal form, transport equation, highly-oscillatory regime, Vlasov equation, strong magnetic field.
Mathematics Subject Classification: Primary:34C29, 82B40, 35Q83.
Citation: Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats. Averaging of highly-oscillatory transport equations. Kinetic & Related Models, 2020, 13 (6) : 1107-1133. doi: 10.3934/krm.2020039
References:
[1]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, A pedagogical approach to the Magnus expansion, European Journal of Physics, 31 (2010), 907-918. doi: 10.1088/0143-0807/31/4/020.  Google Scholar

[2]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field. Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058. doi: 10.1137/070689383.  Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034.  Google Scholar

[5]

M. Bostan, Gyro-kinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957. doi: 10.1137/090777621.  Google Scholar

[6]

P. Chartier, A. Murua and J. M. Sanz-Serna, Erratum to: Higher-order averaging, formal series and numerical integration Ⅱ: The quasi-periodic case, Found Comput Math, 17 (2017), 625-626.  doi: 10.1007/s10208-016-9311-2.  Google Scholar

[7]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found Comput Math, 15, 2015, 591-612. doi: 10.1007/s10208-013-9175-7.  Google Scholar

[8]

P. Chartier, A. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete and Continuous Dynamical Systems (DCDS-A), 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[9]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅱ: The quasi-periodic case, Found Comput Math, 12 (2012), 471-508. doi: 10.1007/s10208-012-9118-8.  Google Scholar

[10]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅰ : B-series, Found Comput Math, 10, 2010, 695-727. doi: 10.1007/s10208-010-9074-0.  Google Scholar

[11]

N. Crouseilles, S. Jin and M. Lemou, Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations, Math. Mod. Meth. App. Sc., 27 (2017), 2031-2070. doi: 10.1142/S0218202517500385.  Google Scholar

[12]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field : Formal derivation, J. Stat. Physics, 165 (2016), 765-784. doi: 10.1007/s10955-016-1645-2.  Google Scholar

[13]

E. Frénod, P.-A. Raviart and E. Sonnendrücker, Two scale expansion of a singularly perturbed convection equation, J. Maths. Pures Appl., 80 (2001), 815-843.  doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[14]

E. Frénod and E. Sonnendrücker, Long time behavior of the Vlasov equation with strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.  Google Scholar

[15]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[17]

A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms, Discrete Mechanics, Geometric Integration and Lie-Butcher Series: DMGILBS, 115-137, Springer Proc. Math. Stat., 267, Springer, Cham, 2018 doi: 10.1007/978-3-030-01397-4_4.  Google Scholar

[18]

A. Murua and J. M. Sanz-Serna, Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Analysis, 138 (2016), 326-345. doi: 10.1016/j.na.2015.10.013.  Google Scholar

[19]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found Comput Math, 17 (2017), 675-712.  doi: 10.1007/s10208-015-9295-3.  Google Scholar

[20]

L. M. Perko, Higher order averaging and related methods for perturbed periodic and quasi-periodic systems, SIAM J. Applied. Math., 17 (1969), 698-724.  doi: 10.1137/0117065.  Google Scholar

[21]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59. Springer-Verlag, 1985. doi: 10.1007/978-1-4757-4575-7.  Google Scholar

show all references

References:
[1]

S. Blanes, F. Casas, J. A. Oteo and J. Ros, A pedagogical approach to the Magnus expansion, European Journal of Physics, 31 (2010), 907-918. doi: 10.1088/0143-0807/31/4/020.  Google Scholar

[2]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field. Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058. doi: 10.1137/070689383.  Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034.  Google Scholar

[5]

M. Bostan, Gyro-kinetic Vlasov equation in three dimensional setting. Second order approximation, SIAM J. Multiscale Model. Simul., 8 (2010), 1923-1957. doi: 10.1137/090777621.  Google Scholar

[6]

P. Chartier, A. Murua and J. M. Sanz-Serna, Erratum to: Higher-order averaging, formal series and numerical integration Ⅱ: The quasi-periodic case, Found Comput Math, 17 (2017), 625-626.  doi: 10.1007/s10208-016-9311-2.  Google Scholar

[7]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅲ: Error bounds, Found Comput Math, 15, 2015, 591-612. doi: 10.1007/s10208-013-9175-7.  Google Scholar

[8]

P. Chartier, A. Murua and J. M. Sanz-Serna, A formal series approach to averaging: Exponentially small error estimates, Discrete and Continuous Dynamical Systems (DCDS-A), 32 (2012), 3009-3027. doi: 10.3934/dcds.2012.32.3009.  Google Scholar

[9]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅱ: The quasi-periodic case, Found Comput Math, 12 (2012), 471-508. doi: 10.1007/s10208-012-9118-8.  Google Scholar

[10]

P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration Ⅰ : B-series, Found Comput Math, 10, 2010, 695-727. doi: 10.1007/s10208-010-9074-0.  Google Scholar

[11]

N. Crouseilles, S. Jin and M. Lemou, Nonlinear Geometric Optics method based multi-scale numerical schemes for highly-oscillatory transport equations, Math. Mod. Meth. App. Sc., 27 (2017), 2031-2070. doi: 10.1142/S0218202517500385.  Google Scholar

[12]

P. Degond and F. Filbet, On the asymptotic limit of the three dimensional Vlasov-Poisson system for large magnetic field : Formal derivation, J. Stat. Physics, 165 (2016), 765-784. doi: 10.1007/s10955-016-1645-2.  Google Scholar

[13]

E. Frénod, P.-A. Raviart and E. Sonnendrücker, Two scale expansion of a singularly perturbed convection equation, J. Maths. Pures Appl., 80 (2001), 815-843.  doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[14]

E. Frénod and E. Sonnendrücker, Long time behavior of the Vlasov equation with strong external magnetic field, Math. Models Methods Appl. Sci., 10 (2000), 539-553.  doi: 10.1142/S021820250000029X.  Google Scholar

[15]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247.  doi: 10.1137/S0036141099364243.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[17]

A. Murua and J. M. Sanz-Serna, Averaging and computing normal forms with word series algorithms, Discrete Mechanics, Geometric Integration and Lie-Butcher Series: DMGILBS, 115-137, Springer Proc. Math. Stat., 267, Springer, Cham, 2018 doi: 10.1007/978-3-030-01397-4_4.  Google Scholar

[18]

A. Murua and J. M. Sanz-Serna, Computing normal forms and formal invariants of dynamical systems by means of word series, Nonlinear Analysis, 138 (2016), 326-345. doi: 10.1016/j.na.2015.10.013.  Google Scholar

[19]

A. Murua and J. M. Sanz-Serna, Word series for dynamical systems and their numerical integrators, Found Comput Math, 17 (2017), 675-712.  doi: 10.1007/s10208-015-9295-3.  Google Scholar

[20]

L. M. Perko, Higher order averaging and related methods for perturbed periodic and quasi-periodic systems, SIAM J. Applied. Math., 17 (1969), 698-724.  doi: 10.1137/0117065.  Google Scholar

[21]

J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sciences, Vol. 59. Springer-Verlag, 1985. doi: 10.1007/978-1-4757-4575-7.  Google Scholar

[1]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[2]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317
[3]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[4]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259
[5]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[6]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[7]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025
[8]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[9]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[10]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[11]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[12]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370
[13]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
[14]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[15]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[16]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079
[17]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081
[18]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364
[19]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020454
[20]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (60)
  • HTML views (116)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences