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A Charge Preserving Scheme for the Numerical Resolution of the Vlasov-Ampère Equations

Published online by Cambridge University Press:  20 August 2015

Nicolas Crouseilles  and
Thomas Respaud
Nicolas Crouseilles*
Affiliation:
INRIA-Nancy-Grand Est, CALVI Project, Strasbourg, France
Thomas Respaud
Affiliation:
IRMA, Universitè de Strasbourg and INRIA-Nancy-Grand Est, CALVI Project, Strasbourg, France
*
*Corresponding author.Email:[email protected]
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Abstract

In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère equation is achieved, with a forward Semi-Lagrangian method introduced in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson’s equation, or equivalently under charge conservation, the Ampère’s one, which self-consistently rules the electric field evolution. In order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-spline deposition will be used for the interpolation step. The solving of the characteristics will be made with a Runge-Kutta 2 method and with a Cauchy-Kovalevsky procedure.

Keywords

65C2085-0868U2065Z05Semi-Lagrangian methodcharge conservationRunge-KuttaCauchy-KovalevskyB-spline deposition
Type
Research Article
Information
Communications in Computational Physics , Volume 10 , Issue 4 , October 2011 , pp. 1001 - 1026
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Barthelmé, R., Le problème de Conservation de la Charge dans le Couplage des Equations de Vlasov et de Maxwell, Thèse de l’Universitè Louis Pasteur, 2005.Google Scholar
[2]Bostan, M. and Crouseilles, N., Convergence of a semi-Lagrangian scheme for the reduced Vlasov-Maxwell sytem for laser-plasma interaction, Numer. Math., 112 (2009), 169–195.Google Scholar
[3]Birdsall, C. K. and Langdon, A. B., Plasma Physics via Computer Simulation, Inst. of Phys. Publishing, Bristol/Philadelphia, 1991.Google Scholar
[4]Bouchut, F., Golse, F. and Pulvirenti, M., Kinetic Equations and Asymptotic Theory, Series in Applied Math, Ciarlet, P. G and Lions, P. L (Eds), Gauthier Villars, 2008.Google Scholar
[5]Carrillo, J. A. and Labrunie, S., Global solutions for the one-dimensional Vlasov-Maxwell system for laser-plasma interaction, Math. Model. Methods. Appl. Sci., 16 (2006), 19–57.CrossRefGoogle Scholar
[6]Carrillo, J.-A. and Vecil, F., Non oscillatory interpolation methods applied to Vlasov-based models, SIAM J. Sci. Comput., 29 (2007), 1179–1206.CrossRefGoogle Scholar
[7]Cheng, C. Z. and Knorr, G., The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22 (1976), 330–3351.Google Scholar
[8]Colella, P. and Woodward, P. R., The Piecewise Parabolic Method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54 (1984), 174–201.Google Scholar
[9]Crouseilles, N., Mehrenberger, M. and Sonnendrücker, E., Conservative semi-Lagrangian schemes for the Vlasov equation, J. Comput. Phys., 229 (2010), 1927–1953.Google Scholar
[10]Crouseilles, N., Respaud, T. and Sonnendrucker, E., A forward semi-Lagrangian method for the numerical solution of the Vlasov equation, Comput. Phys. Commun., 180 (2009), 1730–1745.Google Scholar
[11]Crouseilles, N., Ghizzo, A. and Salmon, S., Vlasov laser-plasma interaction simulations with a moving grid, INRIA Research Report, No. 6109, 2007.Google Scholar
[12]Cotter, C. J., Frank, J. and Reich, S., The remapped particle-mesh semi-Lagrangian advection scheme, Q. J. Meteorol. Soc., 133 (2007), 251–260.Google Scholar
[13]Eastwood, J. W., Virtual particle methods, Comput. Phys. Commun., 64(2) (1991), 252–266.CrossRefGoogle Scholar
[14]Elkina, N. V. and Buchner, J., A new conservative unsplit method for the solution of the Vlasov equation, J. Comput. Phys., 213 (2006), 862–875.Google Scholar
[15]Zh, T.. Esirkepov, Exact charge conservation scheme for Particle-in-Cell simulation with an arbitrary form-factor, Comput. Phys. Commun., 135 (2001), 144–153.Google Scholar
[16]Filbet, F., Sonnendrücker, E. and Bertrand, P., Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172 (2001), 166–187.CrossRefGoogle Scholar
[17]Filbet, F. and Sonnendrücker, E., Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 151 (2003), 247–266.Google Scholar
[18]Ghizzo, A., Bertrand, P., Begue, M. L., Johnston, T. W. and Shoucri, M., A Hilbert-Vlasov code for the study of high-frequency plasma beatwave accelerator, IEEE Trans. Plasma. Sci., 24 (1996), 370–378.Google Scholar
[19]Ghizzo, A., Bertrand, P., Shoucri, M., Johnston, T. W., Fijalkow, E. and Feix, M. R., A Vlasov code for the numerical simulation of stimulated Raman scattering, J. Comput. Phys., 90 (1990), 431–457.CrossRefGoogle Scholar
[20]Grandgirard, V., Brunetti, M., Bertrand, P., Besse, N., Garbet, X., Ghendrih, P., Manfredi, G., Sarrazin, Y., Sauter, O., Sonnendrücker, E., Vaclavik, J. and Villard, L., A drift-kinetic semi-Lagrangian 4D code for ion turbulence simulation, J. Comput. Phys., 217 (2006), 395–423.Google Scholar
[21]Huot, F., Ghizzo, A., Bertrand, P., Sonnendrücker, E. and Coulaud, O., Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov-Maxwell system, J. Comput. Phys., 185 (2003), 512–531.CrossRefGoogle Scholar
[22]Huot, F., Ghizzo, A., Bertrand, P., Sonnendrücker, E. and Coulaud, O., Study of a propagation of ultraintense electromagnetic wave through plasma using semi-Lagrangian Vlasov codes, IEEE Trans. Plasma. Sci., 28 (2000), 1209–1223.CrossRefGoogle Scholar
[23]Johnston, T. W., Bertrand, P., Ghizzo, A., Shoucri, M., Fijalkow, E. and Feix, M. R., Simulated Raman scattering: action evolution and particle trapping via Euler-Vlasov, fluid simulation, Phys. Fluids. B., (1992), 2523–2537.Google Scholar
[24]Marder, B., A method for incorporating Gauss’s law into electromagnetic PIC codes, J. Comput. Phys., 68 (1987), 48–55.Google Scholar
[25]Munz, C.-D., Omnes, P., Schneider, R., Sonnendrücker, E. and Voss, U., Divergence correction techniques for Maxwell solversbased on a hyperbolic model, J. Comput. Phys., 161(2) (2000), 484–511.Google Scholar
[26]Nakamura, T. and Yabe, T., Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Commun., 120 (1999), 122–154.CrossRefGoogle Scholar
[27]Pohn, E., Shoucri, M. and Kamelander, M., Study of the formation of a charge separation at a plasma edge, part I: the numerical integration of a Vlasov equation possessing an invariant, Comput. Phys. Commun., 137 (2001), 380–395.Google Scholar
[28]Reich, S., An explicit and conservative remapping strategy for semi-Lagrangian advection, Atmo. Sci. Lett., 8 (2007), 58–63.Google Scholar
[29]Respaud, T. and Sonnendrücker, E., Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations, HAL: hal-00442957, submitted.Google Scholar
[30]Sircombe, N. J. and Arber, T. D., VALIS: A split-conservative scheme for the relativistic 2D, J. Comput. Phys., 228 (2009), 4773–4788.Google Scholar
[31]Shoucri, M., A two-level implicit scheme for the numerical solution of the linearized vorticity equation, Int. J. Numer. Meth. Eng., 17 (1981), 1525–1538.Google Scholar
[32]Shoucri, M., Nonlinear evolution of the bump-on-tail instability, Phys. Fluids., 22 (1979), 2038–2039.Google Scholar
[33]Schmitz, H. and Grauer, R., Comparison of time splitting and backsubstitution methods for integrating Vlasov equation with magnetic fields, Comput. Phys. Commun., 175 (2006), 86–92.Google Scholar
[34]Sonnendrücker, E., Roche, J., Bertrand, P. and Ghizzo, A., The semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comput. Phys., 149 (1999), 201–220.Google Scholar
[35]Umeda, T., Omura, Y., Tominaga, T. and Matsumoto, H., A new charge conservation method in electromagnetic particle-in-cell simulations, Comput. Phys. Commun., 156(1) (2003), 73–85.Google Scholar
[36]Villasenor, J. and Buneman, O., Rigorous charge conservation for local electromagnetic field solvers, Comput. Phys. Commun., 69 (1992), 306–316.Google Scholar
[37]Zerroukat, M., Wood, N. and Staniforth, A., A monotonic and positive-definite filter for a Semi-Lagrangian inherently conserving and efficient (SLICE) scheme, Q. J. R. Meteorol. Soc., 131 (2005), 2923–2936.Google Scholar
[38]Zerroukat, M., Wood, N. and Staniforth, A., The Parabolic Spline Method (PSM) for conservative transport problems, Int. J. Numer. Meth. Fluids., 51 (2006), 1297–1318.Google Scholar