Structure-Preserving Low Rank Tensor Methods For The Vlasov-Poisson And Vlasov-Maxwell Systems

The main computational challenges of solving the Vlasov systems include the high dimensionality of the phase space, nonlinearity, and inherent conservation properties, among others. This dissertation includes two projects. First, we develop a novel Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov-Maxwell (VM) system, and then we extend our method to two species Vlasov-Poisson (VP) as a continuation of previous work (arXiv:2207.00518). The method takes advantage of the tensor friendly structure of the Vlasov equation and employs the low rank hierarchical Tucker decomposition to approximate the Vlasov solution in high dimensions. Hence, the curse of dimensionality can be mitigated. Furthermore, to realize the LoMaC property, the algorithm simultaneously evolves the conservation laws of mass, momentum, and energy alongside the Vlasov equation using a high order conservative method with the kinetic flux vector splitting. By a conservative orthogonal projection, the low rank solution is guaranteed to have the same macroscopic observables updated from the conservation laws. A collection of numerical tests on the VM and two species VP systems are presented to demonstrate the efficiency and efficacy of the proposed algorithms.

Embargo status: Restricted until 09/2028. To request the author grant access, click on the PDF link to the left.