Vlasov simulation is a method to solve time evolution of a plasma by directly time advancing the distribution function in the position-velocity phase space. Unlike conventional PIC (particle-in-cell)simulations using finite number of particles, the Vlasov simulation is free from thermal (numerical) noise, and thus is advantageous in analyzing fine details of nonlinear plasma phenomena. With this background in mind, we have developed a new Vlasov simulation code (1-d in space, 3-d in the velocity space), in order to study basic properties of nonlinear evolution of magnetohydrodynamic (MHD) waves in the solar wind. In contrast to traditional Vlasov simulations in which electron waves are of major concern, our simulation code focuses on solving plasma behavior around the ion scales, assuming the massless electron fluid. Since we mainly deal with low frequency MHD waves propagating quasi-parallel to the background magnetic field, cyclotron coupling can be assumed to be weak for parameters typical to the solar wind. Thus the Vlasov equation is solved only along the longitudinall direction whereas the MHD equations are solved for the transverse directions. Propagation of Alfven and ion acoustic waves in the simulation is shown to satisfy theoretically obtained dispersion relations. Some results on parametric decay instability of Alfven waves are also presented.
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