Low-frequency electromagnetic (Darwin) applications in plasma simulation

Abstract

Plasma modelers have long sought to be free of the restrictive constraint on discretized time and space representations due to light waves. This constraint, commonly called the CFL condition, implies stability for explicit integration of Maxwell's hyperbolic partial differential equations as long as electromagnetic waves do not propagate more than the smallest grid spacing in a time step. The Darwin limit of Maxwell's equations eliminates these purely electromagnetic modes, making it an effective model for low-frequency phenomena because it retains fidelity for all physics resolved by the large time step that it permits. The early Darwin models suffered from numerical instabilities and non-intuitive boundary conditions. Nielson and Lewis first constructed numerically stable algorithms for the Darwin model but problems associated with vector decomposition remained. Decomposition is expensive and appears to be required in most facets of the electromagnetic calculation. Additionally, decompositions require boundary conditions that are beyond physical intuition. Over the last 15 years, both the physics model and the numerical problem have been significantly extended and restructured. These new formulations eliminate most, if not all, of the vector decomposition; the most demanding questions about boundary conditions do not arise. An overview is given of the most commonly used Darwin models, starting with a brief description of the underlying concept. Several variants of Darwin algorithms are presented; non-neutral and quasi-neutral finite-electron-mass, and quasi-neutral zero-electron-mass embodiments are included. Also discussed are new numerical methods that increase the range of parameters for which these models are practical. Finally a new Darwin variant is described that can follow the time dependence of surface and bulk currents in magnetically active materials (i.e. superconductors). Plasma need to be present-reflecting the new uses that are being found for traditional plasma algorithms.