A Local Macroscopic Conservative (LoMaC) low rank tensor method for the Vlasov dynamics

TL;DR

This paper introduces a new low rank tensor method called LoMaC for simulating the Vlasov-Poisson system, ensuring local conservation of mass, momentum, and energy at the discrete level, improving physical accuracy.

Contribution

The paper develops a novel LoMaC low rank tensor algorithm that guarantees local conservation of mass, momentum, and energy for Vlasov dynamics, extending previous methods to include energy conservation.

Findings

The LoMaC method accurately conserves mass, momentum, and energy in simulations.

Numerical tests demonstrate the method's effectiveness and stability.

The approach efficiently handles high-dimensional problems using hierarchical tensor decompositions.

Abstract

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.