In this paper, we present a new adaptive rank approximation technique for computing solutions to the high-dimensional linear kinetic transport equation. The approach we propose is based on a macro-micro decomposition of the kinetic model in which the angular domain is discretized with a tensor product quadrature rule under the discrete ordinates method. To address the challenges associated with the curse of dimensionality, the proposed low-rank method is cast in the framework of the hierarchical Tucker decomposition. The adaptive rank integrators we propose are built upon high-order discretizations for both time and space. In particular, this work considers implicit-explicit discretizations for time and finite-difference weighted-essentially non-oscillatory discretizations for space. The high-order singular value decomposition is used to perform low-rank truncation of the high-dimensional time-dependent distribution function. The methods are applied to several benchmark problems, where we compare the solution quality and measure compression achieved by the adaptive rank methods against their corresponding full-grid methods. We also demonstrate the benefits of high-order discretizations in the proposed low-rank framework.

翻译：本文提出了一种新的自适应秩近似技术，用于计算高维线性动理学输运方程的解。所提出的方法基于动理学模型的宏微观分解，其中角度域在离散纵标法下采用张量积求积法则进行离散。为应对维度灾难带来的挑战，所提出的低秩方法被置于分层Tucker分解的框架中。我们构建的自适应秩积分器基于时间和空间的高阶离散格式。具体而言，本研究采用隐式-显式格式进行时间离散，并采用有限差分加权本质无振荡格式进行空间离散。通过高阶奇异值分解对高维含时分布函数进行低秩截断。该方法应用于多个基准问题，我们将自适应秩方法的求解质量与压缩效果与其对应的全网格方法进行比较。同时，我们论证了在所提出的低秩框架中采用高阶离散格式的优势。