Projective integration schemes for hyperbolic moment equations

Abstract

In this paper, we apply projective integration methods to hyperbolic moment models of the Boltzmann equation and the BGK equation, and investigate the numerical properties of the resulting scheme. Projective integration is an explicit scheme that is tailored to problems with large spectral gaps between slow and (one or many) fast eigenvalue clusters of the model. The spectral analysis of a linearized moment model clearly shows spectral gaps and reveals the multi-scale nature of the model for which projective integration is a matching choice. The combination of the non-intrusive projective integration method with moment models allows for accurate, but efficient simulations with significant speedup, as demonstrated using several 1D and 2D test cases with different collision terms, collision frequencies and relaxation times.

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps

“…The linearized version of the HME model in Section 2.1 is called Hermite Spectral Method (HSM), see [6,16]. It can be seen as a discrete velocity scheme using a spectral discretization of the velocity space corresponding to variables f i .…”

“…In [16] it was shown that the HSM model contains the same spectrum as a standard discrete velocity model, as commonly used in rarefied gases. This means that the unknown variables relax with relaxation time (x) to an equilibrium manifold on which they only evolve with respect to the macroscopic time scale given by the transport of the macroscopic variables.…”

“…This is true for the BGK operator of the HME model (2.6). For the HSM model (2.9), it requires a previous redefinition of the variable space [16].…”

“…The two beam test case is a standard test case for rarefied gases and was used in [12], [27] for different moment models for constant relaxation time . A spatially varying relaxation time was first tested in [16]. For more detailed information on the test setup, we refer to the literature.…”

“…The relaxation terms in the modeled equations can lead to a time scale separation governed by a small parameter, called the relaxation time [9]. This relaxation time can vary largely throughout the domain, giving rise to different modes developing on different time scales [16].…”