Hybrid (kinetic-fluid) simulation scheme based on method of characteristics

Certain features of the method of characteristics are of considerable interest in relation with Vlasov simulation [H. Abbasi {\it et al}, Phys. Rev. E \textbf{84}, 036702 (2011)]. A Vlasov simulation scheme of this kind can be recurrence free providing initial phase points in velocity space are set randomly. Naturally, less filtering of fine-structures (arising from grid spacing) is possible as there is now a smaller scale than the grid spacing that is average distance between two phase points. Its interpolation scheme is very simple in form and carried out with less operations. In our previous report, the simplest model (immobile ions) was considered to merely demonstrate the important features. Now, a hybrid model is introduced that solves the coupled Vlasov-Fluid-Poisson system self-consistently. A possible application of the code is the study of ion-acoustic (IA) soliton attributes. To this end, a collisionless plasma with hot electrons and cold positive ions is considered. For electrons, the collisionless Vlasov equation is solved by following collisionless phase point trajectories in phase space while ions obey the fluid equations. The periodic boundary conditions are assumed. Both, the characteristic equations of the Vlasov equation and the fluid equations are solved using the Leapfrog-Trapezoidal method. However, to obtain the first half-time step of the Leapfrog, the Euler-Trapezoidal scheme, is employed. The presented scheme conveniently couples the two well-known grids in the Leapfrog method. The first test of the model is an stationary IA soliton. Trapping of electrons is considered and the associated phase space hole is shown. Then as a non-stationary test, the IA soliton generation from a localized initial profile is examined. Conservation laws are the other benchmark tests.