论文标题
弗拉索夫动力学的局部宏观保守（LOMAC）低等级张量法
A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov dynamics
论文作者
论文摘要
在本文中，我们提出了一种新型的局部宏观保守（LOMAC）低等级张量方法，用于模拟Vlasov-Poisson（VP）系统。 LOMAC特性是指在离散水平上对宏观质量，动量和能量的确切局部保护。这是我们先前开发弗拉索夫动力学的保守低等级张量方法的后续工作（ARXIV：2201.10397）。在这项工作中，我们将具有保守奇异值分解（SVD）的低级张量方法应用于高维VP系统，以减轻维度的诅咒，同时保持质量和动量的局部保护。但是，不能保证节能，这是避免非物理等离子体自加热或冷却的关键特性。 LOMAC低等级张量算法中的新成分是，我们同时使用动力学通量载体分开的通量差异形式同时进化了质量，动量和能量的宏观保护定律；然后，通过将低等级动力学解决方案投影到子空间上，该子空间可以实现LOMAC属性，该子空间通过保守的正交投影共享相同的宏观可观察到的子空间。该算法通过溶液张量和相应的保守投影算法的层次结构分解扩展到高维问题。为算法的功效展示了VP系统上的大量数值测试。
In this paper, we propose a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for simulating the Vlasov-Poisson (VP) system. The LoMaC property refers to the exact local conservation of macroscopic mass, momentum and energy at the discrete level. This is a follow-up work of our previous development of a conservative low rank tensor approach for Vlasov dynamics (arXiv:2201.10397). In that work, we applied a low rank tensor method with a conservative singular value decomposition (SVD) to the high dimensional VP system to mitigate the curse of dimensionality, while maintaining the local conservation of mass and momentum. However, energy conservation is not guaranteed, which is a critical property to avoid unphysical plasma self-heating or cooling. The new ingredient in the LoMaC low rank tensor algorithm is that we simultaneously evolve the macroscopic conservation laws of mass, momentum and energy using a flux-difference form with kinetic flux vector splitting; then the LoMaC property is realized by projecting the low rank kinetic solution onto a subspace that shares the same macroscopic observables by a conservative orthogonal projection. The algorithm is extended to the high dimensional problems by hierarchical Tuck decomposition of solution tensors and a corresponding conservative projection algorithm. Extensive numerical tests on the VP system are showcased for the algorithm's efficacy.
