- Open Access
A conservative and non-oscillatory scheme for Vlasov code simulations
Earth, Planets and Space volume 60, pages773–779(2008)
A new numerical positive interpolation technique for conservation laws and its application to Vlasov code simulations are presented. In recent Vlasov simulation codes, the Vlasov equation is solved based on the numerical interpolation method because of its simplicity of algorithm and its ease of programming. However, a large number of grid points are needed in both configuration and velocity spaces to suppress numerical diffusion. In this paper we propose a new high-order interpolation scheme for Vlasov simulations. The current scheme is non-oscillatory and conservative and is well-designed for Vlasov simulations. This is compared with the latest interpolation schemes by performing one-dimensional electrostatic Vlasov simulations.
Arber, T. D. and R. G. L. Vann, A critical comparison of Eulerian-gridbased Vlasov solvers, J. Comput. Phys., 180, 339–357, 2002.
Besse, N. and E. Sonnendrucker, Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space, J. Comput. Phys., 191, 341–376, 2003.
Cheng, C. Z. and G. Knorr, The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22, 330–360, 1976.
Eliasson, B., Numerical modeling of the two-dimensional Fourier transformed Vlasov-Maxwell system J. Comput. Phys., 190, 501–522, 2003.
Elkina, N. V. and J. Buchner, A new conservative unsplit method for the solution of the Vlasov equation, J. Comput. Phys., 213, 862–875, 2005.
Filbet, F. and E. Sonnendrucker, Comparison of Eulerian Vlasov solvers, Comput. Phys. Commun., 150, 247–266, 2003.
Filbet, F., E. Sonnendrucker, and P. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. Phys., 172, 166–187, 2001.
Gagne, R. R. J. and M. M. Shoucri, A splitting scheme for the numerical solution of a one-dimensional Vlasov equation, J. Comput. Phys., 24, 445–449, 1977.
Godunov, S. K., A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations, Math. Sbornik, 47, 271–306, 1959.
Gutnic, M., M. Haefele, I. Pauna, and E. Sonnendrucker, Vlasov simulations on an adaptive phase-space grid, Comput. Phys. Commun., 164, 214–219, 2004.
Klimas, A. J., Numerical method based on the Fourier-Fourier transform approach for modeling 1-D electron plasma evolution, J. Comput. Phys., 50, 270–306, 1983.
Jiang, G.-S. and C.-W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202–228, 1996.
Mangeney, A., F. Califano, C. Cavazzoni, and P. Travnicek, A numerical scheme for the integration of the Vlasov-Maxwell system of equations J. Comput. Phys., 179, 495–538, 2002.
Nakamura, T. and T. Yabe, Cubic interpolated propagation scheme for solving the hyper-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Commun., 120, 122–154, 1999.
Pohn, E., M. Shoucri, and G. Kamelander, Eulerian Vlasov codes, Comput. Phys. Commun., 166, 81–93, 2005.
Ryu, C.-M., T. Rhee, T. Umeda, P. H. Yoon, and Y. Omura, Turbulent acceleration of superthermal electrons, Phys. Plasmas, 14, 100701, 2007.
Schmitz, H. and R. Grauer, Darwin-Vlasov simulations of magnetized plasmas, J. Comput. Phys., 214, 738–756, 2006.
Shoucri, M. and R. R. J. Gagne, Numerical solution of the Vlasov equation by transform methods, J. Comput. Phys., 21, 238–242, 1976.
Sonnendrucker, E., J. Roche, P. Bertrand, and A. Ghizzo, The Semi-Lagrangian method for the numerical resolution of the Vlasov equation, J. Comput. Phys., 149, 201–220, 1999.
Sonnendrucker, E., F. Filbet, A. Friedman, E. Oudet, and J.-L. Vay, Vlasov simulations of beams with amoving grid, Comput. Phys. Commun., 164, 390–395, 2004.
Tanaka, S., T. Umeda, Y. Matsumoto, T. Miyoshi, and T. Ogino, Implementation of non-oscillatory and conservative scheme into magnetohydrodynamic equations, Earth Planets Space, 2008 (under review).
Umeda, T., Vlasov simulation of amplitude-modulated Langmuir waves, Phys. Plasmas, 13, 092304, 2006.
Umeda, T., Vlasov simulation of Langmuir wave packets, Nonlinear Proc. Geophys., 14, 671–679, 2007.
Umeda, T., Y. Omura, P. H. Yoon, R. Gaelzer, and H. Matsumoto, Harmonic Langmuir waves. III. Vlasov simulation, Phys. Plasmas, 10, 382–391, 2003.
Umeda, T., M. Ashour-Abdalla, and D. Schriver, Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code, J. Plasma Phys., 72, 1057–1060, 2006.
Utsumi, T., T. Kunugi, and J. Koga, A numerical method for solving the one-dimensional Vlasov-Poisson equation in phase space, Comput. Phys. Commun., 108, 159–179, 1998.
Xiao, F., T. Yabe, and T. Ito, Constructing oscillation preventing scheme for advection equation by rational function, Comput. Phys. Commun., 93, 1–12, 1999.
Yabe, T., F. Xiao, and T. Utsumi, The constrained interpolation profile method for multiphase analysis, J. Comput. Phys., 169, 556–593, 2001.
Yabe, T., H. Mizoe, K. Takizawa, H. Moriki, H.-N. Im, and Y. Ogata, Higher-order schemes with CIP method and adaptive Soroban grid towards mesh-free scheme, J. Comput. Phys., 194, 57–77, 2004.
Yamamoto, S. and H. Daiguji, Higher-order-accurate upwind schemes for solving the compressible Euler and Navier-Stokes equations, Comput. Fluids, 22, 259–270, 1993.
About this article
Cite this article
Umeda, T. A conservative and non-oscillatory scheme for Vlasov code simulations. Earth Planet Sp 60, 773–779 (2008). https://doi.org/10.1186/BF03352826
- Vlasov equation
- hyperbolic equation
- numerical interpolation
- conservation laws