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Comparison of Eulerian Vlasov solvers. (English) Zbl 1196.82108
Summary: Vlasov methods, which instead of following the particle trajectories, solve directly the Vlasov equation on a grid of phase space have proven to be an efficient alternative to the Particle-In-Cell method for some specific problems. Such methods are useful, in particular, to obtain high precision in regions where the distribution function is small. Gridded Vlasov methods have the advantage of being completely free of numerical noise, however the discrete formulations contain some other numerical artifacts, like diffusion or dissipation. We shall compare in this paper different types of methods for solving the Vlasov equation on a grid in phase space: the semi-Lagrangian method, the finite volume method, the spectral method, and a method based on a finite difference scheme, conserving exactly several invariants of the system. Moreover, for each of those classes of methods, we shall first compare different interpolation or reconstruction procedures. Then we shall investigate the cost in memory as well as in CPU time which is a very important issue because of the size of the problem defined on the phase space.

MSC:
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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[1] Arakawa, A., Computational design for long-term numerical integration of the equation of fluid motion: two-dimensional incompressible flow. part 1, J. comput. phys., J. comput. phys., 135, 1, 103-114, (1997), Reprinted in · Zbl 0939.76068
[2] Boris, J.P.; Book, D.L., Solution of continuity equations by the method of flux-corrected transport, J. comput. phys., 20, 397-431, (1976) · Zbl 0325.76037
[3] Cheng; Knorr, G., J. comput. phys., 22, 330-348, (1976)
[4] Klimas, A.J., A method for overcoming the velocity space filamentation problem in collisionless plasma model solutions, J. comput. phys., 68, 202-226, (1987) · Zbl 0613.76130
[5] Klimas, A.; Farrell, W.M., A splitting algorithm for Vlasov simulation with filamentation filtration, J. comput. phys., 110, 150-163, (1994) · Zbl 0790.76064
[6] Feix, M.R.; Bertrand, P.; Ghizzo, A., Eulerian codes for the Vlasov equation, (), 45 · Zbl 0863.76093
[7] Fijalkow, E., A numerical solution to the Vlasov equation, Comput. phys. comm., 116, 319-328, (1999) · Zbl 1019.76035
[8] Filbet, F.; Sonnendrücker, E.; Bertrand, P., Conservative numerical schemes for the Vlasov equation, J. comput. phys., 172, 166-187, (2001) · Zbl 0998.65138
[9] Ghizzo, A.; Bertrand, P.; Shoucri, M.; Johnston, T.W.; Filjakow, E.; Feix, M.R., A Vlasov code for the numerical simulation of stimulated Raman scattering, J. comput. phys., 90, 431, (1990) · Zbl 0702.76080
[10] Manfredi, G., Long time behavior of non linear Landau damping, Phys. rev. lett., 79, 15, 2815-2818, (1997)
[11] Robert, R.; Sommeria, J., Statistical equilibrium states for two-dimensional flows, J. fluid. mech., 229, 291-310, (1991) · Zbl 0850.76025
[12] Sonnendrücker, E.; Roche, J.; Bertrand, P.; Ghizzo, A., The semi-Lagrangian method for the numerical resolution of Vlasov equations, J. comput. phys., 149, 201-220, (1998) · Zbl 0934.76073
[13] Sonnendrücker, E.; Barnard, J.J.; Friedman, A.; Grote, D.P.; Lund, S.M., Simulation of heavy ion beams with a semi-Lagrangian Vlasov solver, Nucl. instrum. methods in phys. res. A, 464, 1-3, 653-661, (2001)
[14] Nakamura, T.; Yabe, T., Cubic interpolated propagation scheme for solving the hyper-dimensional vlasov – poisson equation in phase space, Comput. phys. comm., 120, 122-154, (1999) · Zbl 1001.82003
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