-
Previous Article
Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder
- KRM Home
- This Issue
-
Next Article
A semigroup approach to the convergence rate of a collisionless gas
Averaging of highly-oscillatory transport equations
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 |
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.
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. |
[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
Other articles
by authors
[Back to Top]