Numerical comparison of an implicit Vlasov–Maxwell solver with macro-particle methods

Assessing the reliability of a macro-particle representation of a kinetic plasma requires understanding its phase-space fidelity as well as conservation properties. While the latter can be expressed analytically, numerical experimentation is required to identify the relative importance of exact conservation as compared with conservation in an average sense for each of the fundamental conservation laws i.e. charge, momentum and energy. Furthermore, the only way to adequately assess the accuracy of the phase-space representation throughout a simulation is by comparison with a demonstrably more accurate method i.e. direct solution of the Vlasov–Maxwell equations. Presently, we take the case of the Weibel instability for our numerical testing. An implicit Vlasov solver in 1-½ dimensions (one spatial and two velocity) is compared to a collection of macro-particle algorithms. These include a standard charge-conserving particle-in-cell code, and variational macro-particle codes using either a spatial grid or a Fourier representation, which provides exact momentum conservation. Finally, we investigate the utility of symplectic integrators for the marco-particle methods.