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.

翻译：稀疏网格方法近年来因其在降低高维动理学方程计算成本方面的潜力而受到关注。本文针对Vlasov-Poisson-Lenard-Bernstein(VPLB)模型，构建了自适应与混合稀疏网格方法。该模型应用于等离子体物理，并在两种简化几何构型中模拟：0x3v空间均匀几何构型和1x3v平板几何构型。我们采用间断伽辽金(DG)方法作为基础离散格式，因其具备高阶精度且能保持偏微分方程的重要结构性质。通过多小波基展开确定稀疏网格基函数与自适应网格判据。我们通过三个典型测试问题对提出的稀疏网格方法进行分析，计算稀疏网格相较于标准DG方法解的节省效果。结果通过自适应稀疏网格离散化库ASGarD获得。