AI 中文总结
针对弗拉索夫 - 麦克斯韦系统，提出张量列车间断伽辽金（TT - DG）公式，结合模态DG离散化与低秩张量表示。经多个基准问题测试，该方法在减少计算成本的同时，保留精度与守恒特性，弱非线性问题压缩比超\(10^4\)，强非线性问题也有效。
AI 中文摘要
我们提出了一种用于弗拉索夫 - 麦克斯韦系统的张量列车间断伽辽金（TT - DG）公式，它将模态DG离散化与相空间解和离散算子的低秩张量表示相结合。该公式利用DG离散化的张量积结构，以压缩形式直接进行求积、微分、非线性迎风格式通量评估和时间积分。在几个标准的1D2V弗拉索夫 - 麦克斯韦基准问题上对该方法进行了评估，包括流动机理不稳定性、弱朗道阻尼和双流不稳定性问题。结果表明，TT公式在大幅减少内存使用和运行时间的同时，再现了底层全网格DG离散化的精度和守恒特性。对于弱非线性问题，压缩比超过\(10^4\)，相对于全网格求解器有显著加速。对于强非线性双流不稳定性问题，尽管由于精细尺度相空间丝状化导致可压缩性降低，但TT公式仍然有效。这些结果表明，张量列车表示为降低基于确定性DG的动力学等离子体模拟的计算成本提供了一种有效方法，同时保留了底层离散化的良好数值特性。
英文摘要
We present a tensor-train discontinuous Galerkin (TT-DG) formulation for the Vlasov--Maxwell system that combines a modal DG discretization with low-rank tensor representations of the phase-space solution and discrete operators. The formulation exploits the tensor-product structure of the DG discretization to perform quadrature, differentiation, nonlinear upwind flux evaluation, and time integration directly in compressed form. The method is evaluated on several standard 1D2V Vlasov--Maxwell benchmark problems, including the streaming Weibel instability, weak Landau damping, and two-stream instability problems. Across these problems, the TT formulation reproduces the accuracy and conservation behavior of the underlying full-grid DG discretization while substantially reducing memory usage and runtime. For weakly nonlinear problems, compression ratios exceeding $10^4$ are obtained together with significant speedups relative to the full-grid solver. For the strongly nonlinear two-stream instability problem, the TT formulation remains effective despite reduced compressibility caused by fine-scale phase-space filamentation. These results demonstrate that tensor-train representations provide an effective approach for reducing the computational cost of deterministic DG-based kinetic plasma simulations while retaining the favorable numerical properties of the underlying discretization.
