Abstract
In this paper we consider the multi-water-bag model for collisionless kinetic equations. The multi-water-bag representation of the statistical distribution function of particles can be viewed as a special class of exact weak solution of the Vlasov equation, allowing to reduce this latter into a set of hydrodynamic equations while keeping its kinetic character. After recalling the link of the multi-water-bag model with kinetic formulation of conservation laws, we derive different multi-water-bag (MWB) models, namely the Poisson-MWB, the quasineutral-MWB and the electromagnetic-MWB models. These models are very promising because they reveal to be very useful for the theory and numerical simulations of laser-plasma and gyrokinetic physics. In this paper we prove some existence and uniqueness results for classical solutions of these different models. We next propose numerical schemes based on Discontinuous Garlerkin methods to solve these equations. We then present some numerical simulations of non linear problems arising in plasma physics for which we know some analytical results.
Highlights
Vlasov equation is a difficult one mainly because of its high dimensionality
The third model is the electromagnetic-multi-water-bag model which is very useful in laser-plasma interaction because it supplies a physical explanation for the formation of low frequency nonlinear coherent structures which are stable in long time, the so-called KEEN (Kinetic Electron Electrostatic Nonlinear) waves [2, 1, 31, 13]
In this paper we have presented multi-water-bag models for collisionless kinetic equations
Summary
Vlasov equation is a difficult one mainly because of its high dimensionality. For each particle species the distribution function f (r, v, t) is defined in a 6D phase space. The concept of entropic solution is not well suited here because the existence of an entropy inequality means that a diffusionlike (or collision-like) process in velocity occurs on the right hand side of the Vlasov equation This observation has been developped in the theory of kinetic formulation of scalar conservation laws [18, 19, 55, 49, 50, 20]. The appearence of a singularity (discontinuous gradients in z due to the Burgers term) is linked to appearance of trapped particles which is characterized by the formation of vortexes and the development of the filamentation process in the phase space In special cases such as the study of nonlinear gyrokinetic turbulence in a cylinder [12], particles dynamic properties [43] imply that the particles are not trapped but only passing
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