Sketches of the implementation of mesh refinement in WarpX with the electrostatic (left) and electromagnetic (right) solvers. In both cases, the charge/current from particles are deposited at the finest levels first, then interpolated recursively to coarser levels. In the electrostatic case, the potential is calculated first at the coarsest level :math:`L_0`, the solution interpolated to the boundaries of the refined patch :math:`r` at the next level :math:`L_{1}` and the potential calculated at :math:`L_1`. The procedure is repeated iteratively up to the highest level. In the electromagnetic case, the fields are computed independently on each grid and patch without interpolation at boundaries. Patches are terminated by absorbing layers (PML) to prevent the reflection of electromagnetic waves. Additional coarse patch :math:`c` and fine grid :math:`a` are needed so that the full solution is obtained by substitution on :math:`a` as :math:`F_{n+1}(a)=F_{n+1}(r)+I[F_n( s )-F_{n+1}( c )]` where :math:`F` is the field, and :math:`I` is a coarse-to-fine interpolation operator. In both cases, the field solution at a given level :math:`L_n` is unaffected by the solution at higher levels :math:`L_{n+1}` and up, allowing for mitigation of some spurious effects (see text) by providing a transition zone via extension of the patches by a few cells beyond the desired refined area (red & orange rectangles) in which the field is interpolated onto particles from the coarser parent level only.
0 commit comments