181 research outputs found
Physics-based adaptivity of a spectral method for the Vlasov-Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space
We propose a spectral method for the 1D-1V Vlasov-Poisson system where the
discretization in velocity space is based on asymmetrically-weighted Hermite
functions, dynamically adapted via a scaling and shifting of the
velocity variable. Specifically, at each time instant an adaptivity criterion
selects new values of and based on the numerical solution of the
discrete Vlasov-Poisson system obtained at that time step. Once the new values
of the Hermite parameters and are fixed, the Hermite expansion is
updated and the discrete system is further evolved for the next time step. The
procedure is applied iteratively over the desired temporal interval. The key
aspects of the adaptive algorithm are: the map between approximation spaces
associated with different values of the Hermite parameters that preserves total
mass, momentum and energy; and the adaptivity criterion to update and
based on physics considerations relating the Hermite parameters to the
average velocity and temperature of each plasma species. For the discretization
of the spatial coordinate, we rely on Fourier functions and use the implicit
midpoint rule for time stepping. The resulting numerical method possesses
intrinsically the property of fluid-kinetic coupling, where the low-order terms
of the expansion are akin to the fluid moments of a macroscopic description of
the plasma, while kinetic physics is retained by adding more spectral terms.
Moreover, the scheme features conservation of total mass, momentum and energy
associated in the discrete, for periodic boundary conditions. A set of
numerical experiments confirms that the adaptive method outperforms the
non-adaptive one in terms of accuracy and stability of the numerical solution
Recurrence phenomenon for Vlasov-Poisson simulations on regular finite element mesh
In this paper, we focus on one difficulty arising in the numerical simulationof the Vlasov-Poisson system: when using a regular grid-based solver with periodicboundary conditions, perturbations present at the initial time artificially reappear at alater time. For regular finite-element mesh in velocity, we show that this recurrence timeis actually linked to the spectral accuracy of the velocity quadrature when computingthe charge density. In particular, choosing trigonometric quadrature weights optimallydefers the occurence of the recurrence phenomenon. Numerical results using the Semi-Lagrangian Discontinuous Galerkin and the Finite Element / Semi-Lagrangian methodconfirm the analysis
The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations
We discuss the development, analysis, implementation, and numerical
assessment of a spectral method for the numerical simulation of the
three-dimensional Vlasov-Maxwell equations. The method is based on a spectral
expansion of the velocity space with the asymmetrically weighted Hermite
functions. The resulting system of time-dependent nonlinear equations is
discretized by the discontinuous Galerkin (DG) method in space and by the
method of lines for the time integration using explicit Runge-Kutta
integrators. The resulting code, called Spectral Plasma Solver (SPS-DG), is
successfully applied to standard plasma physics benchmarks to demonstrate its
accuracy, robustness, and parallel scalability
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2018
This open access book features a selection of high-quality papers from the presentations at the International Conference on Spectral and High-Order Methods 2018, offering an overview of the depth and breadth of the activities within this important research area. The carefully reviewed papers provide a snapshot of the state of the art, while the extensive bibliography helps initiate new research directions
- …