Particle-In-Cell techniques on non-uniform meshes

The Particle-In-Cell (PIC) technique is the most popular algorithmic choice for the solution of the kinetic equations in plasma physics. In PIC, computational particles, each representing a small element of phase space, move through a computational mesh where the field equations responsible for inter-particle interactions are solved. It is the interplay between the particles and the mesh that makes the PIC algorithm very efficient, because the cost of the algorithm scales linearly with the number of particles N (as opposed to N-body/molecular-dynamics-type simulations where the cost of the algorithm is proportional to N²). Traditionally, the great majority of PIC methods have been developed on uniform meshes and with explicit time discretizations. This means that the method must resolve the shortest scales and fastest frequencies to maintain numerical stability.



Several application domains, however, demand PIC algorithms on non-uniform meshes. An example is the interaction of plasmas and material objects, such as a spacecraft moving in the near-Earth space plasma environment, where objects bring new spatial and temporal scales into the problem which could be much shorter than the relevant plasma scales and could render the application of a uniform-mesh PIC code unfeasible. Thus, PIC methods on non-uniform meshes have been developed with a variety of approaches (unstructured, adaptive-mesh-refinement and multi-block structured meshes). A common problem to all non-uniform PIC codes, however, is that of the self-force, i.e. the fact that a computational particle exerts an unphysical force on itself as a result of the interplay of computing the fields on a mesh and interpolating the force field back to the particles. In this talk, we will review the most common non-uniform PIC approaches, discuss their advantages and disadvantages and strategies for mitigation of the self-force, and conclude with some areas of needed future developments.