A Charge Conserving Exponential Predictor Corrector FEMPIC Formulation for Relativistic Particle Simulations

Abstract

The state of art of charge-conserving electromagnetic finite element particle-in-cell has grown by leaps and bounds in the past few years. These advances have primarily been achieved for leap-frog time stepping schemes for Maxwell solvers, in large part, due to the method strictly following the proper space for representing fields, charges, and measuring currents. Unfortunately, leap-frog based solvers (and their other incarnations) are only conditionally stable. Recent advances have made Electromagnetic Finite Element Particle-in-Cell (EM-FEMPIC) methods built around unconditionally stable time stepping schemes were shown to conserve charge. Together with the use of a quasi-Helmholtz decomposition, these methods were both unconditionally stable and satisfied Gauss' Laws to machine precision. However, this architecture was developed for systems with explicit particle integrators where fields and velocities were off by a time step. While completely self-consistent methods exist in the literature, they follow the classic rubric: collect a system of first order differential equations (Maxwell and Newton equations) and use an integrator to solve the combined system. These methods suffer from the same side-effect as earlier--they are conditionally stable. Here we propose a different approach; we pair an unconditionally stable Maxwell solver to an exponential predictor-corrector method for Newton's equations. As we will show via numerical experiments, the proposed method conserves energy within a PIC scheme, has an unconditionally stable EM solve, solves Newton's equations to much higher accuracy than a traditional Boris solver and conserves charge to machine precision. We further demonstrate benefits compared to other polynomial methods to solve Newton's equations, like the well known Boris push.

arXiv comment12 pages, 15 figures

References in corpus (10)

• The Energy Conserving Particle-in-Cell Method
• Geometric Finite Element Discretization of Maxwell Equations in Primal and Dual Spaces
• Exact Charge-Conserving Scatter-Gather Algorithm for Particle-in-Cell Simulations on Unstructured Grids: A Geometric Perspective
• On the Boris solver in particle-in-cell simulation
• Structure-Preserving Geometric Particle-in-Cell Methods for Vlasov-Maxwell Systems
• Energy-conserving time propagation for a geometric particle-in-cell Vlasov--Maxwell solver
• A Set of Benchmark Tests for Validation of 3D Particle In Cell Methods
• Time Integrator Agnostic Charge Conserving Finite Element PIC
• Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell
• An Envelope Tracking Approach for Particle in Cell Simulations