An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

Abstract

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method.