Semi-Lagrangian Discontinuous Galerkin Method with Adaptive Mesh Refinement for the Vlasov--Poisson System in 1X+3V

TL;DR

This paper introduces an advanced semi-Lagrangian discontinuous Galerkin method with adaptive mesh refinement for solving the Vlasov--Poisson system in 1X+3V, improving efficiency and accuracy in high-dimensional plasma simulations.

Contribution

It extends the SLDG method to adaptive meshes and three-dimensional velocity space, developing hybrid strategies for conforming and nonconforming cells and implementing tensor-product DG elements.

Findings

Demonstrates correct Landau damping rates

Achieves exact mass conservation

Shows convergence with polynomial degree and AMR levels

Abstract

We extend the semi-Lagrangian discontinuous Galerkin (SLDG) method of Einkemmer to velocity grids with adaptive mesh refinement (AMR) and to three-dimensional velocity space. The original SLDG formulation assumes uniform cell widths, which permits the overlap matrices to be precomputed once per fractional shift and reused for every cell. On an adaptively refined mesh, neighboring cells may differ in size, invalidating this assumption. We develop a hybrid sweep strategy: conforming cells in the mesh interior use precomputed per-level overlap matrices (the fast path), while nonconforming cells at refinement boundaries evaluate generalized overlap integrals on the fly (the slow path). A compressed sparse row (CSR) pencil data structure organizes the dimensional splitting along each velocity coordinate, with weighted accumulation for coarse cells that appear in multiple pencils. The method is extended from one to three velocity dimensions using tensor-product DG elements on hexahedral cells provided by PETSc's PetscFE class. We verify the solver on the standard Landau damping benchmark in 1X+3V, demonstrating correct damping rates, exact mass conservation, and convergence behavior with polynomial degree and AMR refinement level.