摘要：In this work, 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.

个人简介： 熊涛，中国科学技术大学数学科学学院教授，国家高层次青年人才，主要从事计算流体力学和动理学方程的高效高精度数值方法的研究。近些年来，主要围绕多尺度动理学方程的渐近保持算法展开研究，发展了高效高阶的渐近保持间断Galerkin有限元方法等。目前主持国家重大研究计划重点培育项目，参与科技部重点专项项目等。曾主持国家自然科学面上基金、国防科研核科学挑战专题子课题等项目。