A physics-based-preconditioned quasi-Newton solver for energy- and momentum-conserving implicit hybrid particle-in-cell simulations

Abstract

Explicit hybrid particle-in-cell (PIC) schemes, with kinetic ions and fluid electrons, accumulate numerical heating over long simulations and are unstable to finite-grid instabilities. Energy-conserving implicit integration removes heating and instabilities, but may be inefficient due to the need of nonlinear iteration. We present a fast quasi-Newton solver for an implicit hybrid-PIC formulation. The nonlinear residual employs the electric field and electron pressure as nonlinear unknowns, and nonlinearly eliminates particles. The formulation is strictly energy conserving and, on Cartesian meshes, also momentum conserving. The nonlinear iteration is efficiently performed with a quasi-Newton solver that replaces the true Jacobian with an approximate one without spoiling conservation properties upon convergence. The approximate Jacobian replaces the particle ion linear response with an analytical mass matrix that is local and requires minimal storage. This approximate Jacobian is efficiently inverted with a physics-based multigrid preconditioner, adapted from Hall MHD. We verify the scheme linearly with whistler, kinetic Alfvén, and cylindrical normal modes, and nonlinearly with the GEM reconnection challenge. On GEM, explicit integration needs a forty-times-smaller time step, and nine times the wall-clock time, to reach an energy conservation comparable to that of the implicit solver. We apply the method to the simulation of a three-dimensional field-reversed configuration formation by external coils.