# American Institute of Mathematical Sciences

June  2011, 4(2): 441-477. doi: 10.3934/krm.2011.4.441

## An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits

 1 INRIA-Nancy Grand Est and IRMA, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France 2 CNRS and IRMAR, Université de Rennes 1, 263 Avenue du General Leclerc CS74205, 35042 Rennes cedex, France

Received  October 2010 Revised  February 2011 Published  April 2011

In this work, we extend the micro-macro decomposition based numerical schemes developed in [3] to the collisional Vlasov-Poisson model in the diffusion and high-field asymptotics. In doing so, we first write the Vlasov-Poisson model as a system that couples the macroscopic (equilibrium) part with the remainder part. A suitable discretization of this micro-macro model enables to derive an asymptotic preserving scheme in the diffusion and high-field asymptotics. In addition, two main improvements are presented: On the one hand a self-consistent electric field is introduced, which induces a specific discretization in the velocity direction, and represents a wide range of applications in plasma physics. On the other hand, as suggested in [30], we introduce a suitable reformulation of the micro-macro scheme which leads to an asymptotic preserving property with the following property: It degenerates into an implicit scheme for the diffusion limit model when $\varepsilon\rightarrow 0$, which makes it free from the usual diffusion constraint $\Delta t=O(\Delta x^2)$ in all regimes. Numerical examples are used to demonstrate the efficiency and the applicability of the schemes for both regimes.
Citation: Nicolas Crouseilles, Mohammed Lemou. An asymptotic preserving scheme based on a micro-macro decomposition for Collisional Vlasov equations: diffusion and high-field scaling limits. Kinetic & Related Models, 2011, 4 (2) : 441-477. doi: 10.3934/krm.2011.4.441
##### References:

show all references

##### References:
 [1] Anaïs Crestetto, Nicolas Crouseilles, Mohammed Lemou. Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles. Kinetic & Related Models, 2012, 5 (4) : 787-816. doi: 10.3934/krm.2012.5.787 [2] Vincent Giovangigli, Wen-An Yong. Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion. Kinetic & Related Models, 2015, 8 (1) : 79-116. doi: 10.3934/krm.2015.8.79 [3] Vincent Giovangigli, Wen-An Yong. Erratum: Volume viscosity and internal energy relaxation: Symmetrization and Chapman-Enskog expansion''. Kinetic & Related Models, 2016, 9 (4) : 813-813. doi: 10.3934/krm.2016018 [4] Naoufel Ben Abdallah, Hédia Chaker. Mixed high field and diffusion asymptotics for the fermionic Boltzmann equation. Kinetic & Related Models, 2009, 2 (3) : 403-424. doi: 10.3934/krm.2009.2.403 [5] Viktor I. Gerasimenko, Igor V. Gapyak. Hard sphere dynamics and the Enskog equation. Kinetic & Related Models, 2012, 5 (3) : 459-484. doi: 10.3934/krm.2012.5.459 [6] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 [7] Céline Baranger, Marzia Bisi, Stéphane Brull, Laurent Desvillettes. On the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic rarefied gases. Kinetic & Related Models, 2018, 11 (4) : 821-858. doi: 10.3934/krm.2018033 [8] Seok-Bae Yun. Entropy production for ellipsoidal BGK model of the Boltzmann equation. Kinetic & Related Models, 2016, 9 (3) : 605-619. doi: 10.3934/krm.2016009 [9] Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205 [10] Francesca Marcellini. Free-congested and micro-macro descriptions of traffic flow. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 543-556. doi: 10.3934/dcdss.2014.7.543 [11] Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019 [12] Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019146 [13] Carmen Cortázar, Manuel Elgueta, Fernando Quirós, Noemí Wolanski. Asymptotic behavior for a nonlocal diffusion equation on the half line. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1391-1407. doi: 10.3934/dcds.2015.35.1391 [14] Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393 [15] Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic & Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036 [16] Daomin Cao, Hang Li. High energy solutions of the Choquard equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3023-3032. doi: 10.3934/dcds.2018129 [17] Jan Haskovec, Nader Masmoudi, Christian Schmeiser, Mohamed Lazhar Tayeb. The Spherical Harmonics Expansion model coupled to the Poisson equation. Kinetic & Related Models, 2011, 4 (4) : 1063-1079. doi: 10.3934/krm.2011.4.1063 [18] Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 [19] Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590 [20] Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093

2018 Impact Factor: 1.38