A Nodal Discontinuous Galerkin Method with Low-Rank Velocity Space Representation for the Multi-Scale BGK Model

TL;DR

This paper introduces a hybrid low-rank velocity space method combined with a nodal discontinuous Galerkin approach to efficiently solve the multi-scale BGK model, enabling accurate and robust simulations of gas kinetics.

Contribution

It presents a novel low-rank velocity decomposition integrated with a nodal discontinuous Galerkin method for the BGK model, extending low-rank techniques to realistic multi-scale kinetic problems.

Findings

Demonstrates high-order accuracy in canonical tests

Reduces computational complexity significantly

Shows robustness across various multi-scale regimes

Abstract

A novel hybrid algorithm is presented for the Boltzmann-BGK equation, in which a low-rank decomposition is applied solely in the velocity subspace, while a full-rank representation is maintained in the physical (position) space. This approach establishes a foundation for extending modern low-rank techniques to solve the Boltzmann equation in realistic settings, particularly where structured representations -- such as conformal geometries -- may not be feasible in practical engineering applications. A nodal discontinuous Galerkin method is employed for spatial discretization, coupled with a low-rank decomposition over the velocity grid, as well as implicit-explicit Runge-Kutta methods for time integration. To handle the limit of vanishing collision time, a multiscale implicit integrator based on an auxiliary moment equation is utilized. The algorithm's order of accuracy, reduced computational complexity, and robustness are demonstrated on a suite of canonical gas kinetics problems with increasing complexity.