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

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.

Figures (6)

• Figure 3.1: SLDG sweep geometry on an AMR mesh (1D cross-section). The mesh has a coarse cell $L$ (width $2h$, level $\ell$) and two fine cells $R_1$, $R_2$ (width $h$, level $\ell{+}1$). The red dot (right) marks the target IP $v^*$; the black dot marks the foot point $\bar{v} = v^* - a\,\Delta t$. The brace shows the foot interval. Shaded rectangles indicate the fast-path (crosshatch, same-level source) and slow-path (NE hatching, cross-level source) contributions to the $L^2$ projection.
• Figure 3.2: 2D SLDG sweep geometry on an AMR mesh. Coarse cell $L$ (width $2h$) and fine cells $R_1$, $R_2$ (width $h$). Upper band (Sweep A): coarse-cell IP sweeps rightward into $R_2$. Lower band (Sweep B): fine-cell IP sweeps leftward into $L$. Crosshatch = fast path (same level); NE hatching = slow path (cross level). The vertical extent of the bands is schematic; the overlap integrals are 1D (see Fig. \ref{['fig:sldg-amr-geometry']}).
• Figure 3.3: 2D cross-sections ($v_x$--$v_y$ plane at $v_z = 0$) of the electron distribution $u_e$ on two AMR velocity meshes ($R=6$). (a) The $3^3$ base mesh with one AMR level; the four inner base cells are each refined once, concentrating resolution near the Maxwellian peak. (b) The $4^3$ base mesh with one AMR level; the finer base grid resolves the Maxwellian shoulders more accurately. (Visit uses linear interpolation from vertices)
• Figure 6.1: Damping rate error vs. integration points. Left: h-convergence on uniform meshes ($L=0$) for $p=3,4,5$; each point is labeled with the base grid size $N_b^3$. Right: AMR efficiency --- uniform meshes (solid, faded) vs. AMR meshes (dashed, triangles) at each polynomial degree. All runs: $R=6$, $N_x=64$, $\Delta t=0.1$, $t_{\rm end}=20$.
• Figure 6.2: Left: $E_{\max}$ vs. time (semilogy) for selected configurations; the black dashed line shows the envelope fit and black triangles mark detected peaks for the best configuration (Q5, $4^3$ uniform). Right top: relative mass conservation error $|\Delta m_0|/|m_0(0)|$. Right bottom: relative total-energy conservation error $|\Delta E_{\rm tot}|/|E_{\rm tot}(0)|$. All runs: $R=6$, $N_x=64$, $\Delta t=0.1$, $t_{\rm end}=20$.