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A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps
arXiv - CS - Numerical Analysis Pub Date : 2021-06-16 , DOI: arxiv-2106.08834
Wei Guo, Jing-Mei Qiu

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 is 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.

中文翻译:

线性传输和非线性 Vlasov 解的低秩张量表示及其相关流图

我们提出了一种低秩张量方法来近似线性传输和非线性 Vlasov 解及其相关的流图。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,我们提出了一种通过从 PDE 的离散化中添加新的基函数来动态和自适应地建立低秩解基的新方法,并且从 SVD 类型的截断过程中删除基础。对于离散化,我们采用高阶有限差分空间离散化和二阶强稳定性保持多步时间离散化。我们应用相同的程序以低秩的方式演化流图的动态,当流图享有低秩结构时,这被证明是有利的,而溶液具有高阶或显示丝状结构。对于高维问题,采用分层塔克分解。执行了一组广泛的线性和非线性 Vlasov 测试示例,以显示该算法的高阶空间和时间收敛性,网格细化高达 SVD 类型的截断,所提出的低秩方法的显着计算节省,尤其是对于高维问题,改进了具有细丝的解决方案的流程图方法的性能。
更新日期:2021-06-17
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