American Institute of Mathematical Sciences

Advanced
October  2016, 9(5): 1447-1473. doi: 10.3934/dcdss.2016058

Expansion of a singularly perturbed equation with a two-scale converging convection term

Alexandre Mouton 1,

1. 

Laboratoire Paul Painlevé, CNRS & Universit é de Sciences et Technologies Lille 1, Cit é Scientifique, F-59655 Villeneuve d'Ascq, France

Received  February 2015 Revised  September 2015 Published  October 2016

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.
Keywords: homogenization., Vlasov equation, convection equation, two-scale convergence, gyrokinetic approximations.
Mathematics Subject Classification: Primary: 35Q83, 76M40, 78A35, 82D10; Secondary: 76M4.
Citation: Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  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]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New-York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[5]

A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. Google Scholar

[6]

A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[7]

P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Model. Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278.  Google Scholar

[8]

D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.  Google Scholar

[9]

F. Filbet and É. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Model. Meth. Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[10]

E. Frénod, M. Gutnic and S. Hirstoaga, First order two-scale particle-in-cell numerical method for the Vlasov equation, ESAIM Proc., 38 (2012), 348-360. doi: 10.1051/proc/201238019.  Google Scholar

[11]

E. Frnod and A. Mouton, Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar

[12]

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

[13]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar

[14]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.  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]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.  Google Scholar

[18]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ESAIM Math. Model. Numer. Anal., 46 (2012), 929-947. doi: 10.1051/m2an/2011068.  Google Scholar

[19]

D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574.  Google Scholar

[20]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.  Google Scholar

[21]

E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85. doi: 10.1007/BF02547774.  Google Scholar

[22]

H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309.  Google Scholar

[23]

W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. Google Scholar

[24]

W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. Google Scholar

[25]

R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053.  Google Scholar

[26]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009.  Google Scholar

[27]

A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251.  Google Scholar

[28]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[29]

K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182.  Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  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]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New-York, 2013. doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[5]

A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. Google Scholar

[6]

A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[7]

P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Model. Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278.  Google Scholar

[8]

D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.  Google Scholar

[9]

F. Filbet and É. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Model. Meth. Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340.  Google Scholar

[10]

E. Frénod, M. Gutnic and S. Hirstoaga, First order two-scale particle-in-cell numerical method for the Vlasov equation, ESAIM Proc., 38 (2012), 348-360. doi: 10.1051/proc/201238019.  Google Scholar

[11]

E. Frnod and A. Mouton, Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar

[12]

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

[13]

E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar

[14]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213.  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]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.  Google Scholar

[18]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ESAIM Math. Model. Numer. Anal., 46 (2012), 929-947. doi: 10.1051/m2an/2011068.  Google Scholar

[19]

D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574.  Google Scholar

[20]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33.  Google Scholar

[21]

E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85. doi: 10.1007/BF02547774.  Google Scholar

[22]

H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309.  Google Scholar

[23]

W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. Google Scholar

[24]

W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. Google Scholar

[25]

R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053.  Google Scholar

[26]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009.  Google Scholar

[27]

A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251.  Google Scholar

[28]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

[29]

K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182.  Google Scholar

[1]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[2]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[3]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[4]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[5]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[6]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[7]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[8]

Iryna Pankratova, Andrey Piatnitski. Homogenization of convection-diffusion equation in infinite cylinder. Networks & Heterogeneous Media, 2011, 6 (1) : 111-126. doi: 10.3934/nhm.2011.6.111
[9]

Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827
[10]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[11]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[12]

Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190
[13]

Patrick Henning. Convergence of MsFEM approximations for elliptic, non-periodic homogenization problems. Networks & Heterogeneous Media, 2012, 7 (3) : 503-524. doi: 10.3934/nhm.2012.7.503
[14]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[15]

Marina Chugunova, Dmitry Pelinovsky. Two-pulse solutions in the fifth-order KdV equation: Rigorous theory and numerical approximations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 773-800. doi: 10.3934/dcdsb.2007.8.773
[16]

Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations & Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070
[17]

Vitali Vougalter, Vitaly Volpert. On the solvability conditions for the diffusion equation with convection terms. Communications on Pure & Applied Analysis, 2012, 11 (1) : 365-373. doi: 10.3934/cpaa.2012.11.365
[18]

Adrien Dekkers, Anna Rozanova-Pierrat, Vladimir Khodygo. Models of nonlinear acoustics viewed as approximations of the Kuznetsov equation. Discrete & Continuous Dynamical Systems, 2020, 40 (7) : 4231-4258. doi: 10.3934/dcds.2020179
[19]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[20]

Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic & Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (74)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences