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Sparse-grid Discontinuous Galerkin Methods for the Vlasov-Poisson-Lenard-Bernstein Model

Stefan Schnake, Coleman Kendrick, Eirik Endeve, Miroslav Stoyanov, Steven Hahn, Cory D Hauck, David L Green, Phil Snyder, John Canik

TL;DR

The paper develops adaptive sparse-grid discontinuous Galerkin methods for the VPLB kinetic model in slab geometries, achieving substantial memory savings while maintaining accuracy. It introduces a hierarchical multiwavelet basis, a sparse-grid selection rule, and adaptive refinement/coarsening strategies, implemented in the ASGarD library. Through numerical experiments on relaxation, Riemann, and collisional Landau damping problems, the adaptive sparse-grid method demonstrates strong performance relative to full-grid and mixed-grid approaches, especially for velocity-space resolution and higher-order moments. The results indicate that adaptive sparse grids effectively mitigate the curse of dimensionality for high-dimensional kinetic problems and highlight directions for extending to full 3x3v simulations and improved preservation properties.

Abstract

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov-Poisson-Lenard-Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

Sparse-grid Discontinuous Galerkin Methods for the Vlasov-Poisson-Lenard-Bernstein Model

TL;DR

The paper develops adaptive sparse-grid discontinuous Galerkin methods for the VPLB kinetic model in slab geometries, achieving substantial memory savings while maintaining accuracy. It introduces a hierarchical multiwavelet basis, a sparse-grid selection rule, and adaptive refinement/coarsening strategies, implemented in the ASGarD library. Through numerical experiments on relaxation, Riemann, and collisional Landau damping problems, the adaptive sparse-grid method demonstrates strong performance relative to full-grid and mixed-grid approaches, especially for velocity-space resolution and higher-order moments. The results indicate that adaptive sparse grids effectively mitigate the curse of dimensionality for high-dimensional kinetic problems and highlight directions for extending to full 3x3v simulations and improved preservation properties.

Abstract

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov-Poisson-Lenard-Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.
Paper Structure (25 sections, 2 theorems, 63 equations, 15 figures, 2 algorithms)

This paper contains 25 sections, 2 theorems, 63 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Vlasov--Poisson--Lenard--Bernstein Model
  3. Geometric reductions
  4. Space homogeneous problem
  5. Reduction to slab geometry
  6. Reduction of 1x1v
  7. Notation and the Discontinuous Galerkin Method
  8. Notation
  9. Discontinuous Galerkin Method
  10. Time Stepping Method
  11. Sparse-grid Method
  12. Single Dimension Wavelet Basis
  13. Multiwavelets
  14. The Sparse-grid Selection Rule
  15. Adaptive Sparse-grids
  16. ...and 10 more sections

Key Result

Proposition 1

The LB operator satisfies the following properties

Figures (15)

  • Figure 4.1.1: Plots of the wavelet basis $g_{\ell,j}^i$, given by \ref{['eqn:wavelet_basis_1D_def']}, for $k=2$. In each plot, the entire set of wavelet basis functions for level $\ell=3$ and lower are shown in each plot and are translucent.
  • Figure 4.3.1: Sparse-grid illustrations.
  • Figure 4.4.1: Riemann problem -- \ref{['subsec:riemann']} -- $\nu=1$: Adaptive Sparse-grid Method at $t=0.04918$. The threshold is $\tau=10^{-4}$ and the adaptive sparse-grid cannot refine past $\bm{\ell}=(7,6,6,6)$.
  • Figure 5.2.1: Relaxation Problem -- \ref{['subsec:relax']}: 2D plot of the velocity distribution $f_h(v_x,v_y,v_z=0.019)$ at the start (left) and end (right) of a relaxation simulation. These results were obtained with a full-grid run with $\bm{\ell}=(5,5,5)$.
  • Figure 5.2.2: Relaxation Problem -- \ref{['subsec:relax']}: Plots of interest for full-grid runs with varying levels.
  • ...and 10 more figures

Theorems & Definitions (6)

  • Proposition 1: endeveHauck_2022
  • Proposition 2: endeveHauck_2022
  • Definition 1
  • Definition 2: wang2016sparseBungartz_Griebel_2004
  • Definition 3
  • Definition 4