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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  December 2020 Early access  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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

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