By edited by H. Usui and Y. Omura.

It is a number of prolonged lecture notes of the tutorials given on the foreign tuition for house Simulations (ISSS)-7, March 2005, by means of the invited academics who've been actively desirous about laptop simulation thoughts in house plasma physics.

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1 Semi-Lagrange methods Given the above difficulties with fully Lagrangian methods semi-Lagrange schemes have been developed. Such schemes first use the fact that the most accurate way to solve convection (or advection, term vδr f ) hyperbolic PDE is to use their characteristics, along which the function value (here the distribution function) stays constant. The characteristics of the Vlasov-equation are the particle orbits in response to the selfconsistently calculated electromagnetic forces (this is also used in PIC codes).

Collisionless damping of nonlinear plasma oscillations, Phys. Fluids, 8(12), 2255–2262, 1965. , C. E. Watt, R. B. Horne, and M. P. Freeman, Anomalous resistivity in non-Maxwellian plasmas, J. Geophys. , 108, 1442–1453, 2003. , A spectral algorithm for solving the relativistic Vlasov-Maxwell equations, preprint NASA/TP2001-210195, 19 pages, 2001. Sigov, Yu. , Computer simulation of plasma turbulence in open systems, Physica Scripta, 2(2), 367, 1982. Silin, I. and J. B¨uchner, Nonlinear instability of thin current sheets in antiparallel and guided magnetic fields, Phys.

B¨uchner, A new conservative unsplit method for the solution of the Vlasov equation, J. Comput. , 213(2), 862–875, 2006. , Numerical solution of the Vlasov-equation: the 1D code, Comput. Phys. , 116, 329– 335, 1999. 46 J. , E. Sonnendr¨ucker, and N. Bertrand, Conservative numerical schemes for the Vlasov equation, J. Comput. , 172, 166–187, 2001. Gagn´e, R. and M. Shoucri, A splitting scheme for the numerical solution of a one-dimensional Vlasovequation, J. Comput. , 24, 445, 1977. Gary, S. , Theory of Space Plasma Microinstabilities, Cambridge University Press, Cambridge, UK, pp.