Journal of Computational Physics
A finite element code for the simulation of one-dimensional Vlasov plasmas I. Theory
Journal of Computational Physics
A finite element code for the simulation of one-dimensional Vlasov plasmas. II.Applications
Journal of Computational Physics
Spectral methods on triangles and other domains
Journal of Scientific Computing
A splitting algorithm for Vlasov simulation with filamentation filtration
Journal of Computational Physics
The semi-Lagrangian method for the numerical resolution of the Vlasov equation
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Conservative numerical schemes for the Vlasov equation
Journal of Computational Physics
Plasma Physics Via Computer
Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems
SIAM Journal on Numerical Analysis
A critical comparison of Eulerian-grid-based Vlasov solvers
Journal of Computational Physics
Numerical modelling of the two-dimensional Fourier transformed Vlasov-Maxwell system
Journal of Computational Physics
Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space
Journal of Computational Physics
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications
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This paper presents the development of parallel direct Vlasov solvers with discontinuous Galerkin (DG) method for beam and plasma simulations in four dimensions. Both physical and velocity spaces are in two dimesions (2P2V) with unstructured mesh. Contrary to the standard particle-in-cell (PIC) approach for kinetic space plasma simulations, i.e., solving Vlasov-Maxwell equations, direct method has been used in this paper. There are several benefits to solving a Vlasov equation directly, such as avoiding noise associated with a finite number of particles and the capability to capture fine structure in the plasma. The most challanging part of a direct Vlasov solver comes from higher dimensions, as the computational cost increases as $N^{2d}$, where $d$ is the dimension of the physical space. Recently, due to the fast development of supercomputers, the possibility has become more realistic. Many efforts have been made to solve Vlasov equations in low dimensions before; now more interest has focused on higher dimensions. Different numerical methods have been tried so far, such as the finite difference method, Fourier Spectral method, finite volume method, and spectral element method. This paper is based on our previous efforts to use the DG method. The DG method has been proven to be very successful in solving Maxwell equations, and this paper is our first effort in applying the DG method to Vlasov equations. DG has shown several advantages, such as local mass matrix, strong stability, and easy parallelization. These are particularly suitable for Vlasov equations. Domain decomposition in high dimensions has been used for parallelization; these include a highly scalable parallel two-dimensional Poisson solver. Benchmark results have been shown and simulation results will be reported.