A sparse-grids filter for structure-preserving electromagnetic PIC methods*

In the past decade, the idea of using the sparse-grid recombination technique as a means to reduce noise in electrostatic particle-in-cell (PIC) methods has generated substantial interest. Likewise, structure-preserving electromagnetic (EM) PIC methods based on discretizing the Lagrangian or Hamiltonian structure of the Vlasov-Maxwell system have been a prominent direction of inquiry for their ability to automatically conserve known invariants of the continuous dynamics (e.g. energy and Gauss's laws). This work unites these two ideas to provide a structure-preserving EM PIC method with a noise-reducing sparse-grids filter. Specially designed filter matrices, which are sparse operators with a convenient Kronecker structure, are inserted into the discrete Lagrangian of a general variational electromagnetic PIC method in such a manner that the symmetries of the Lagrangian are preserved. This yields a filtered, variational EM PIC method which retains all the structure-preserving properties of a usual variational EM PIC method, which enjoys the noise-reduction properties of a sparse-grids PIC method, and whose implementation involves only a small, inexpensive modification of existing structure-preserving EM PIC methods. The strategy is general and should be amenable to a broad class of variational EM PIC methods. Numerical examples are provided using the GEMPIC method.