Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system

Abstract

In this paper, we propose energy-conserving numerical schemes for the Vlasov-Ampère (VA) systems. The VA system is a model used to describe the evolution of probability density function of charged particles under self consistent electric field in plasmas. It conserves many physical quantities, including the total energy which is comprised of the kinetic and electric energy. Unlike the total particle number conservation, the total energy conservation is challenging to achieve. For simulations in longer time ranges, negligence of this fact could cause unphysical results, such as plasma self heating or cooling. In this paper, we develop the first Eulerian solvers that can preserve fully discrete total energy conservation. The main components of our solvers include explicit or implicit energy-conserving temporal discretizations, an energy-conserving operator splitting for the VA equation and discontinuous Galerkin finite element methods for the spatial discretizations. We validate our schemes by rigorous derivations and benchmark numerical examples such as Landau damping, two-stream instability and bump-on-tail instability.

Cited by in corpus (24)

• GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods
• Vlasov methods in space physics and astrophysics
• Discontinuous Galerkin algorithms for fully kinetic plasmas
• High Order Maximum Principle Preserving Semi-Lagrangian Finite Difference WENO schemes for the Vlasov Equation
• Energy-conserving Discontinuous Galerkin Methods for the Vlasov-Maxwell System
• A High Order Time Splitting Method Based on Integral Deferred Correction for Semi-Lagrangian Vlasov Simulations
• Metriplectic Integrators for the Landau Collision Operator
• A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting
• Quadratic conservative scheme for relativistic Vlasov--Maxwell system
• Arbitrarily high order Convected Scheme solution of the Vlasov-Poisson system
• Micro-macro decomposition based asymptotic-preserving numerical schemes and numerical moments conservation for collisional nonlinear kinetic equations
• The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations
• Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma
• Highly efficient energy-conserving moment method for the multi-dimensional Vlasov-Maxwell system
• An efficient, conservative, time-implicit solver for the fully kinetic arbitrary-species 1D-2V Vlasov-Ampère system
• Hamiltonian Particle-in-Cell methods for Vlasov-Poisson equations
• An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method for Vlasov-Maxwell Equations
• Numerical Study of the Two-Species Vlasov-Ampère System: Energy-Conserving Schemes and the Current-Driven Ion-Acoustic Instability
• Conservative DG Method for the Micro-Macro Decomposition of the Vlasov-Poisson-Lenard-Bernstein Model
• Discontinuous Galerkin schemes for a class of Hamiltonian evolution equations with applications to plasma fluid and kinetic problems
• Energy conserving particle-in-cell methods for relativistic Vlasov--Maxwell equations of laser-plasma interaction
• Sparse Grid Discontinuous Galerkin Methods for the Vlasov-Maxwell System
• A sparse grid discontinuous Galerkin method for high-dimensional transport equations and its application to kinetic simulations
• All-speed numerical methods for the Euler equations via a sequential explicit time integration