A Nodal Discontinuous Galerkin Method with Rank-Adaptive Velocity Space Representation for the Multiscale BGK Model

Abstract

A novel hybrid algorithm is presented for the Boltzmann-BGK equation, in which a rank-adaptive 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 rank-adaptive 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 rank-adaptive 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.