Discontinuous Galerkin Methods for the Vlasov--Maxwell Equations

Abstract

Abstract. Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Maxwell system. The proposed scheme employs discontinuous Galerkin discretizations for both the Vlasov and the Maxwell equations, resulting in a consistent description of the distribution function and electromagnetic fields. It is proven, up to some boundary effects, that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the Maxwell equations and the underlying approximation spaces. Error estimates are established for several flux choices. The scheme is tested on the streaming Weibel instability: the order of accuracy and conservation properties of the proposed method are verified.

A Kinetic Vlasov Model for Plasma Simulation Using Discontinuous Galerkin Method on Many-Core Architectures

“…The discontinuous Galerkin method is selected for its robustness to work well for kinetic equations in phase space, Maxwell's electromagnetic equations, and fluid equations in physical space with multiple contemporary examples for these application areas [12,13,14,15,16,17,18]. Additionally, the explicit method is straightforward to implement and well suited for emerging high performance computing architectures as described in Chapters 4 and 9.…”

“…(3.12) is developed here as a compact linear algebra system per-node and per-element to evaluate the time derivative . 13) where the geometric Jacobian J k = ∂r ∂x…”

“…[13] was reproduced as one means to validate the WARPM Vlasov-Maxwell model implementation in (1D + 2V ).…”

A posteriori error estimates for the one and one-half Dimensional Relativistic Vlasov–Maxwell system

“…For SD studies relevant to our approach see, e.g., [1,2] and the references therein some containing also further studies involving discontinuous Galerkin schemes and their developments. A priori error estimates for a discontinuous Galerkin approach, that is not based on SD framework, are derived in [8]. The study of [8] relies on an appropriate choice of numerical flux and, unlike our fully discrete scheme, is split into separate spatial and temporal discretizations.…”

“…A priori error estimates for a discontinuous Galerkin approach, that is not based on SD framework, are derived in [8]. The study of [8] relies on an appropriate choice of numerical flux and, unlike our fully discrete scheme, is split into separate spatial and temporal discretizations.…”