论文标题
弗拉索夫动力学的保守低等级张量方法
A conservative low rank tensor method for the Vlasov dynamics
论文作者
论文摘要
在本文中，我们提出了一种保守的低等级张量方法来近似非线性vlasov解决方案。低级方法是基于我们早期的工作（ARXIV：2106.08834）。它利用了这一事实，即弗拉索夫方程中的差分运算符对张量友好，我们建议通过添加微分方程的离散化，并从奇异值分解（SVD）-Type否定过程中删除基础来动态和适应性地构建低级解决方案基础。为了进行离散化，我们采用高阶有限差空间离散以及二阶强稳定性保留多步骤时间离散化。 尽管SVD截断将消除代表高维弗拉索夫解决方案的冗余，但它将破坏相关的全保守方案的保护特性。在本文中，我们开发了一种保守的截断程序，并保存质量，动量和动能密度。保守的截断是通过对$ 1 $，$ V $和$ v^2 $跨越与加权内部产品相关的速度空间中的$ 1 $，$ v $和$ v^2 $的子空间的正交投影实现的。然后，该算法对剩余的加权SVD截断进行，涉及缩放，然后进行标准的SVD截断和重新缩放。该算法在高维的高维张量分解上进一步开发了高维vlasov溶液的分解，从而克服了维数的诅咒。进行了大量的非线性弗拉索夫示例，以显示拟议的保守低级方法的有效性和保护性。对质量，动量和能量的保护史的非保守低等级张量方法进行了比较。
In this paper, we propose a conservative low rank tensor method to approximate nonlinear Vlasov solutions. The low rank approach is based on our earlier work (arxiv: 2106.08834). It takes advantage of the fact that the differential operators in the Vlasov equation are tensor friendly, based on which we propose to dynamically and adaptively build up low rank solution basis by adding new basis functions from discretization of the differential equation, and removing basis from a singular value decomposition (SVD)-type truncation procedure. For the discretization, we adopt a high order finite difference spatial discretization together with a second order strong stability preserving multi-step time discretization. While the SVD truncation will remove the redundancy in representing the high dimensional Vlasov solution, it will destroy the conservation properties of the associated full conservative scheme. In this paper, we develop a conservative truncation procedure with conservation of mass, momentum and kinetic energy densities. The conservative truncation is achieved by an orthogonal projection onto a subspace spanned by $1$, $v$ and $v^2$ in the velocity space associated with a weighted inner product. Then the algorithm performs a weighted SVD truncation of the remainder, which involves a scaling, followed by the standard SVD truncation and rescaling back. The algorithm is further developed in high dimensions with hierarchical Tucker tensor decomposition of high dimensional Vlasov solutions, overcoming the curse of dimensionality. An extensive set of nonlinear Vlasov examples are performed to show the effectiveness and conservation property of proposed conservative low rank approach. Comparison is performed against the non-conservative low rank tensor approach on conservation history of mass, momentum and energy.
