F. Aràndiga, A. Baeza, A. M. Belda, and P. Mulet, Analysis of WENO Schemes for Full and Global Accuracy, SIAM Journal on Numerical Analysis, vol.49, issue.2, pp.893-915, 2011.
DOI : 10.1137/100791579

A. B. Navarro, P. Morel, M. Albrecht-marc, D. Carati, F. Merz et al., Free energy cascade in gyrokinetic turbulence, Physical Review Letters, pp.106-055001, 2011.

A. J. Brizard and T. Hahm, Foundations of nonlinear gyrokinetic theory, Reviews of modern physics, pp.421-468, 2007.

C. Cercignani, The Boltzmann equation and its applications, 1988.
DOI : 10.1007/978-1-4612-1039-9

C. Z. Cheng and G. Knorr, The integration of the vlasov equation in configuration space, Journal of Computational Physics, vol.22, issue.3, pp.330-351, 1976.
DOI : 10.1016/0021-9991(76)90053-X

P. Colella, M. R. Dorr, J. A. Hittinger, and D. F. Martin, High-order, finite-volume methods in mapped coordinates, Journal of Computational Physics, vol.230, issue.8, pp.2952-2976, 2011.
DOI : 10.1016/j.jcp.2010.12.044

N. Crouseilles, A. Ratnani, and E. Sonnendrücker, An Isogeometric Analysis approach for the study of the gyrokinetic quasi-neutrality equation, Journal of Computational Physics, vol.231, issue.2, pp.231-373, 2012.
DOI : 10.1016/j.jcp.2011.09.004
URL : https://hal.archives-ouvertes.fr/inria-00584672

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, 1998.

M. R. Feix, P. Bertrand, and A. Ghizzo, EULERIAN CODES FOR THE VLASOV EQUATION, Series on Advances in Mathematics for Applied Sciences Kinetic Theory and Computing, pp.45-81, 1994.
DOI : 10.1142/9789814354165_0002

F. Filbet and E. Sonnendrückder, Comparison of Eulerian Vlasov solvers, Computer Physics Communications, vol.150, issue.3, pp.247-266, 2003.
DOI : 10.1016/S0010-4655(02)00694-X
URL : https://hal.archives-ouvertes.fr/hal-00129663

F. Filbet and C. Yang, An inverse Lax???Wendroff method for boundary conditions applied to Boltzmann type models, Journal of Computational Physics, vol.245, pp.43-61, 2013.
DOI : 10.1016/j.jcp.2013.03.015
URL : https://hal.archives-ouvertes.fr/hal-00732107

F. Filbet and C. Yang, Numerical Simulations of Kinetic Models for Chemotaxis, SIAM Journal on Scientific Computing, vol.36, issue.3, pp.348-366, 2014.
DOI : 10.1137/130910208
URL : https://hal.archives-ouvertes.fr/hal-00798822

A. Ghizzo, P. Bertrand, M. Shoucri, T. W. Johnston, E. Filjakow et al., A Vlasov code for the numerical simulation of stimulated Raman scattering, Journal of Computional Physics, pp.90-431, 1990.

X. Garbet, Y. Idomura, L. Villard, and T. H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, vol.50, issue.4, pp.50-043002, 2010.
DOI : 10.1088/0029-5515/50/4/043002

V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet et al., A drift-kinetic Semi-Lagrangian 4D code for ion turbulence simulation, Journal of Computational Physics, vol.217, issue.2, pp.217-395, 2006.
DOI : 10.1016/j.jcp.2006.01.023
URL : https://hal.archives-ouvertes.fr/hal-00594856

V. Grandgirard, Y. Sarazin, X. Garbet, G. Dif-pradalier, . Ph et al., Computing ITG turbulence with a full-f semi-Lagrangian code, Communications in Nonlinear Science and Numerical Simulation, vol.13, issue.1, pp.13-81, 2008.
DOI : 10.1016/j.cnsns.2007.05.016
URL : https://hal.archives-ouvertes.fr/hal-00141419

A. Herten and S. Osher, Uniformly High-Order Accurate Nonoscillatory Schemes. I, SIAM Journal on Numerical Analysis, vol.24, issue.2, pp.279-309, 1987.
DOI : 10.1137/0724022

G. Jiang and C. Shu, Efficient Implementation of Weighted ENO Schemes, Journal of Computational Physics, vol.126, issue.1, pp.202-228, 1996.
DOI : 10.1006/jcph.1996.0130

R. L. Miller, M. S. Chu, J. M. Greene, Y. R. Lin-liu, and R. E. Waltz, Noncircular, finite aspect ratio, local equilibrium model, Physical Plasmas, pp.973-978, 1998.

J. Pétri, Non-linear evolution of the diocotron instability in a pulsar electrosphere: two-dimensional particle-in-cell simulations, Astronomy and Astrophysics, vol.503, issue.1, pp.1-12, 2009.
DOI : 10.1051/0004-6361/200911778

R. Samtaney, estimates of sub-grid-scale terms, Computational Science & Discovery, vol.5, issue.1, p.14004, 2012.
DOI : 10.1088/1749-4699/5/1/014004

G. Strang, On the Construction and Comparison of Difference Schemes, SIAM Journal on Numerical Analysis, vol.5, issue.3, pp.506-517, 1968.
DOI : 10.1137/0705041

E. Sonnendrücker and J. Roche, The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation, Journal of Computational Physics, vol.149, issue.2, pp.201-220, 1999.
DOI : 10.1006/jcph.1998.6148

S. Tan and C. Shu, Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws, Journal of Computational Physics, vol.229, issue.21, pp.8144-8166, 2010.
DOI : 10.1016/j.jcp.2010.07.014

C. Yang and F. Filbet, Conservative and non-conservative methods based on Hermite weighted essentially non-oscillatory reconstruction for Vlasov equations, Journal of Computational Physics, vol.279
DOI : 10.1016/j.jcp.2014.08.048
URL : https://hal.archives-ouvertes.fr/hal-00918077

&. Université-de-lyon, I. C. Inria, and . Jordan, 43 boulevard 11 novembre 1918, F-69622 Villeurbanne cedex, FRANCE E-mail address, F. Filbet: filbet@math.univ-lyon1, 92 West Dazhi Street, Nan Gang District, 150001.