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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.
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.
|
[3] |
M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics,, J. Differential Equations, 249 (2010), 1620.
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 (2013).
doi: 10.1007/978-1-4614-5975-0. |
[5] |
A. Brizard, Nonlinear Gyrokinetic Tokamak Physics,, Ph.D thesis, (1990). Google Scholar |
[6] |
A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.
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.
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.
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.
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.
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.
|
[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.
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.
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.
|
[15] |
E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227.
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.
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.
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.
doi: 10.1051/m2an/2011068. |
[19] |
D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.
doi: 10.1137/090774574. |
[20] |
D. Han-Kwan, The three-dimensional finite Larmor radius approximation,, Asymptot. Anal., 66 (2010), 9.
|
[21] |
E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen,, Acta Math., 58 (1932), 57.
doi: 10.1007/BF02547774. |
[22] |
H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations,, Michigan Math. J., 9 (1962), 303.
|
[23] |
W.-W. Lee, Gyrokinetic approach in particle simulation,, Phys. Fluids, 26 (1983), 555. Google Scholar |
[24] |
W.-W. Lee, Gyrokinetic particle simulation model,, J. Comp. Phys., 72 (1987), 243. Google Scholar |
[25] |
R.-G. Littlejohn, A guiding center Hamiltonian: A new approach,, J. Math. Phys., 20 (1979), 2445.
doi: 10.1063/1.524053. |
[26] |
A. Mouton, Approximation Multi-échelles de L'équation de Vlasov,, Ph.D thesis, (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.
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.
doi: 10.1137/0520043. |
[29] |
K. Schmitt, Periodic Solutions of Nonlinear Differential Systems,, J. Math. Anal. Appl., 40 (1972), 174.
|
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.
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.
|
[3] |
M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics,, J. Differential Equations, 249 (2010), 1620.
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 (2013).
doi: 10.1007/978-1-4614-5975-0. |
[5] |
A. Brizard, Nonlinear Gyrokinetic Tokamak Physics,, Ph.D thesis, (1990). Google Scholar |
[6] |
A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.
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.
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.
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.
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.
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.
|
[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.
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.
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.
|
[15] |
E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227.
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.
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.
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.
doi: 10.1051/m2an/2011068. |
[19] |
D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.
doi: 10.1137/090774574. |
[20] |
D. Han-Kwan, The three-dimensional finite Larmor radius approximation,, Asymptot. Anal., 66 (2010), 9.
|
[21] |
E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen,, Acta Math., 58 (1932), 57.
doi: 10.1007/BF02547774. |
[22] |
H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations,, Michigan Math. J., 9 (1962), 303.
|
[23] |
W.-W. Lee, Gyrokinetic approach in particle simulation,, Phys. Fluids, 26 (1983), 555. Google Scholar |
[24] |
W.-W. Lee, Gyrokinetic particle simulation model,, J. Comp. Phys., 72 (1987), 243. Google Scholar |
[25] |
R.-G. Littlejohn, A guiding center Hamiltonian: A new approach,, J. Math. Phys., 20 (1979), 2445.
doi: 10.1063/1.524053. |
[26] |
A. Mouton, Approximation Multi-échelles de L'équation de Vlasov,, Ph.D thesis, (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.
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.
doi: 10.1137/0520043. |
[29] |
K. Schmitt, Periodic Solutions of Nonlinear Differential Systems,, J. Math. Anal. Appl., 40 (1972), 174.
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