A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps
A low rank tensor representation of linear transport and nonlinear Vlasov solutions and their associated flow maps
复制标题
线性传输和非线性 Vlasov 解及其相关流图的低秩张量表示
DOI：
10.1016/j.jcp.2022.111089
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发表时间：
2022
影响因子：
4.1
通讯作者：
Qiu, Jing-Mei
中科院分区：
文献类型：
--
作者：
Guo, Wei;Qiu, Jing-Mei
我们提出了一个低秩张量方法来近似线性传输和非线性Vlasov解决方案及其相关的流图。该方法利用了Vlasov方程中的微分算子是张量友好的这一事实，在此基础上，我们提出了一种新的方法，通过从PDE离散化中添加新的基函数，并从SVD类型的截断过程中删除基，来动态地自适应地建立低秩解基。对于离散，我们采用高阶有限差分空间离散和二阶强稳定性保持多步时间离散。我们应用相同的过程来发展的动态流图在低秩的方式，这被证明是有利的，当流图享有低秩结构，而解决方案遭受高秩或显示reconstruction结构。对高维问题采用分层Tucker分解。一组广泛的线性和非线性Vlasov测试的例子进行显示高阶空间和时间收敛的网格细化算法的SVD型截断，显着的计算节省所提出的低秩的方法，特别是对于高维问题，改进的性能流图方法的解决方案的演示。
We propose a low-rank tensor approach to approximate linear transport and nonlinear Vlasov solutions and their associated flow maps. The approach takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose a novel way to dynamically and adaptively build up low-rank solution basis by adding new basis functions from discretization of the PDE, and removing basis from an SVD-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization and a second order strong stability preserving multi-step time discretization. We apply the same procedure to evolve the dynamics of the flow map in a low-rank fashion, which proves to be advantageous when the flow map enjoys the low rank structure, while the solution suffers from high rank or displays filamentation structures. Hierarchical Tucker decomposition is adopted for high dimensional problems. An extensive set of linear and nonlinear Vlasov test examples are performed to show the high order spatial and temporal convergence of the algorithm with mesh refinement up to SVD-type truncation, the significant computational savings of the proposed low-rank approach especially for high dimensional problems, the improved performance of the flow map approach for solutions with filamentations.