自适应稀疏网格间断伽辽金近似 Bhatnagar--Gross--Krook 模型

中文摘要 AI 辅助

本文研究了 Bhatnagar--Gross--Krook（BGK）模型的自适应稀疏网格间断伽辽金（DG）离散化，该模型是一个在四维和六维相空间中提出的动力学方程。标准 DG 方法因维数灾难而无法实际应用于 BGK 模型，这促使人们采用能随时间适应解的压缩表示。利用自适应稀疏网格 DG 方法，我们通过将自适应自由度与全网格 DG 方法进行比较，并评估在流体和稀薄流态下得到的动力学量和流体量，来量化精度和压缩效果。测试案例包括松弛问题、多维 Sod 激波管以及先前低秩 BGK 研究中使用的剪切/膨胀流。为了在不违反守恒的前提下构建高效的麦克斯韦分布评估——这是保持结构的 BGK 模拟中的一个核心障碍——我们引入了一种混合插值策略，该策略利用速度可分离性来恢复正确的离散碰撞不变量，并证明了所得离散碰撞算子在自适应稀疏网格上的守恒性。我们的结果表明，自适应稀疏网格策略能够恢复具有尖锐梯度的准确且物理相关的解，并且该方法将活跃自由度减少了数倍到数个数量级，其中最大的缩减出现在六维示例中。所有计算均使用开源 ASGarD 自适应稀疏网格 DG 库执行。

英文摘要

This work studies adaptive sparse-grid discontinuous Galerkin (DG) discretizations for the Bhatnagar--Gross--Krook (BGK) model, a kinetic equation posed in four- and six-dimensional phase-space. Standard DG methods are rendered impractical for the BGK model by the curse of dimensionality, motivating compressed representations that adapt to the solution in time. Using the adaptive sparse-grid DG method, we quantify accuracy and compression by comparing the adaptive degrees of freedom to full-grid DG methods and by assessing the resulting kinetic and fluid quantities in both fluid and rarefied regimes. Test cases include a relaxation problem, a multidimensional Sod shock tube, and shear/expansion flows used in prior low-rank BGK studies. To build an efficient Maxwellian evaluation without violating conservation, a central obstacle for structure-perserving BGK simulations, we introduce a hybrid interpolation strategy that exploits velocity separability to recover the correct discrete collision invariants and prove conservation of the resulting discrete collision operator on adaptive sparse grids. Our results show that the adaptive sparse-grid strategy can recover accurate and physically relevant solutions with sharp gradients, and the method reduces the active degrees of freedom by factors ranging from several-fold to several orders of magnitude, with the largest reductions occurring in the six-dimensional examples. All computations are performed with the open-source ASGarD adaptive sparse-grid DG library.

发表机构

• Oak Ridge National Laboratory（橡树岭国家实验室）
• The University of Tennessee（田纳西大学）

机构由 AI 辅助整理，请以论文原文为准。