Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

Abstract

Development of particle in cell methods using finite element based methods (FEMs) have been a topic of renewed interest; this has largely been driven by (a) the ability of finite element methods to better model geometry, (b) better understanding of function spaces that are necessary to represent all Maxwell quantities, and (c) more recently, the fundamental rubrics that should be obeyed in space and time so as to satisfy Gauss' laws and the equation of continuity. In that vein, methods have been developed recently that satisfy these equations and are agnostic to time stepping methods. While is development is indeed a significant advance, it should be noted that implicit FEM transient solvers support an underlying null space that corresponds to a gradient of a scalar potential $\nabla \Phi (\text{r})$ (or $t\nabla \Phi (\text{r})$ in the case of wave equation solvers). While explicit schemes do not suffer from this drawback, they are only conditionally stable, time step sizes are mesh dependent, and very small. A way to overcome this bottleneck, and indeed, satisfy all four Maxwell's equation is to use a quasi-Helmholtz formulation on a tessellation. In the re-formulation presented, we strictly satisfy the equation of continuity and Gauss' laws for both the electric and magnetic flux densities. Results demonstrating the efficacy of this scheme will be presented.

The following article has been submitted to AIP Physics of Plasmas

References in corpus (4)

• Energy-conserving time propagation for a geometric particle-in-cell Vlasov--Maxwell solver
• Explicit high-order noncanonical symplectic algorithms for ideal two-fluid systems
• Relativistic Extension of a Charge-Conservative Finite Element Solver for Time-Dependent Maxwell-Vlasov Equations
• Subcycling of particle orbits in variational, geometric electromagnetic particle-in-cell methods

Cited by in corpus (5)

• Rubrics for Charge Conserving Current Mapping in Finite Element Particle in Cell Methods
• An Asymptotic-Preserving and Energy-Conserving Particle-In-Cell Method for Vlasov-Maxwell Equations
• A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations
• Port Parameter Extraction Based Self Consistent Coupled EM-Circuit FEM Solvers
• Higher Order Charge Conserving Electromagnetic Finite Element Particle in Cell Method