Embedded grid refinement for semi-Lagrangian parallelized relativistic Vlasov–Maxwell solver

We present a new dynamic embedded grid refinement method performed only on the momentum coordinates, which we applied to a relativistic Vlasov–Maxwell model. This refinement strategy, which maintains a uniform space coordinate grid, differs from the one usually adopted in particle in cell (PIC) code. It involves embedded rectangles and may evolve over time. From an algorithmic point of view, it displays the further advantage of allowing one to preserve MPI/OpenMP parallelization strategies possibly implemented in equivalent Vlasov semi-Lagrangian solvers on an Eulerian grid, although it requires a change in the boundary conditions of the interpolation algorithm and in the computing of integrals. A comparison between two versions of the same code, with and without mesh refinement, is here shown for a transition between the oblique Weibel-type instability to the current filamentation instability for 2D2V and 2D3V geometries in both nonrelativistic and relativistic regimes. The mesh refinement method is perfectly adapted to follow the formation of thin filaments in the momentum space associated with the energy transfer in the Weibel-like instabilities and allows a remarkable computational gain.