Direct Vlasov Solvers with High-Order Spectral Element Method (Communications in Computational Physics, 2010)
This paper presents the development of parallel direct Vlasov solvers usingthe Spectral Element Method (SEM). Instead of the standard Particle-In-Cell (PIC)approach for kinetic space plasma simulation, i.e. solving the Vlasov-Maxwell equations,the direct method has been used in this paper. There are several benefits tosolve the Vlasov equation directly, such as avoiding noise associated with the finitenumber of particles and the capability to capture the fine structure in the plasma, etc.The most challenging part of direct Vlasov solver comes from high dimension, as thecomputational cost increases as N2d, where d is the dimension of the physical space.Recently due to fast development of supercomputers, the possibility of high dimensionsbecomes more realistic. A significant effort has been devoted to solve the Vlasovequation in low dimensions so far, now more interests focus on higher dimensions.Different numerical methods have been tried so far, such as finite difference method,Fourier spectral method, finite volume method, etc. In this paper SEM has been successfullyapplied to construct these solvers. SEM has shown several advantages, suchas easy interpolation due to local element structure and long time integration due toits high order accuracy. Domain decomposition in high dimensions have been used forparallelization, these include scalable parallel 1D and 2D Poisson solvers. Benchmarkresults have been shown and simulation results have been reported for two differentcases: one dimension (1P1V), and two dimensions (2P2V) in both physical and velocityspaces.