Log In Sign Up

Energy-conserving explicit and implicit time integration methods for the multi-dimensional Hermite-DG discretization of the Vlasov-Maxwell equations

by   Cecilia Pagliantini, et al.

We study the conservation properties of the Hermite-discontinuous Galerkin (Hermite-DG) approximation of the Vlasov-Maxwell equations. In this semi-discrete formulation, the total mass is preserved independently for every plasma species. Further, an energy invariant exists if central numerical fluxes are used in the DG approximation of Maxwell's equations, while a dissipative term is present when upwind fluxes are employed. In general, traditional temporal integrators might fail to preserve invariants associated with conservation laws (at the continuous or semi-discrete level) during the time evolution. Hence, we analyze the capability of explicit and implicit Runge-Kutta (RK) temporal integrators to preserve such invariants. Since explicit RK methods can only ensure preservation of linear invariants but do not provide any control on the system energy, we develop a novel class of nonlinear explicit RK schemes. The proposed methods can be tuned to preserve the energy invariant at the continuous or semi-discrete level, a distinction that is important when upwind fluxes are used in the discretization of Maxwell's equations since upwind provides a numerical source of energy dissipation that is not present when central fluxes are used. We prove that the proposed methods are able to preserve the energy invariant and to maintain the semi-discrete energy dissipation (if present) according to the discretization of Maxwell's equations. An extensive set of numerical experiments corroborates the theoretical findings. It also suggests that maintaining the semi-discrete energy dissipation when upwind fluxes are used leads to an overall better accuracy of the method relative to using upwind fluxes while forcing exact energy conservation.


page 26

page 27

page 29


Functional equivariance and conservation laws in numerical integration

Preservation of linear and quadratic invariants by numerical integrators...

On Energy Laws and Stability of Runge–Kutta Methods for Linear Seminegative Problems

This paper presents a systematic theoretical framework to derive the ene...

The rotating rigid body model based on a non-twisting frame

This work proposes and investigates a new model of the rotating rigid bo...

Some continuous and discontinuous Galerkin methods and structure preservation for incompressible flows

In this paper, we present consistent and inconsistent discontinuous Gale...