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Asymptotic behaviors of solutions for finite difference analogue of the Chipot-Weissler equation
Expansion of a singularly perturbed equation with a two-scale converging convection term
1. | Laboratoire Paul Painlevé, CNRS & Universit é de Sciences et Technologies Lille 1, Cit é Scientifique, F-59655 Villeneuve d'Ascq, France |
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. |
[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] |
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. |
[5] |
A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[19] |
D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367.
doi: 10.1137/090774574. |
[20] |
D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33. |
[21] |
E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85.
doi: 10.1007/BF02547774. |
[22] |
H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309. |
[23] |
W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. |
[24] |
W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. |
[25] |
R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458.
doi: 10.1063/1.524053. |
[26] |
A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009. |
[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. |
[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. |
[29] |
K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182. |
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[2] |
M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. |
[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] |
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. |
[5] |
A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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] |
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. |
[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. |
[19] |
D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367.
doi: 10.1137/090774574. |
[20] |
D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33. |
[21] |
E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85.
doi: 10.1007/BF02547774. |
[22] |
H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309. |
[23] |
W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. |
[24] |
W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. |
[25] |
R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458.
doi: 10.1063/1.524053. |
[26] |
A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009. |
[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. |
[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. |
[29] |
K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182. |
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