  In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale
environment. The model equations can be regarded as a hyperbolic regularization
and are derived from kinetic equations. So far, the research on QGD models has
been focused on problems with constant coefficients. In this paper, we
investigate the QGD model in multiscale media, which can be used in porous
media applications. This multiscale problem is interesting from a multiscale
methodology point of view as the model problem has a hyperbolic multiscale
term, and designing multiscale methods for hyperbolic equations is challenging.
In the paper, we apply the constraint energy minimizing generalized multiscale
finite element method (CEM-GMsFEM) combined with the leapfrog scheme in time to
solve this problem. The CEM-GMsFEM provides a flexible and systematical
framework to construct crucial multiscale basis functions for approximating the
solution to the problem with reduced computational cost. With this approach of
spatial discretization, we establish the stability of the fully discretized
scheme under a relaxed version of the so-called CFL condition. Complete
convergence analysis of the proposed method is presented. Numerical results are
provided to illustrate and verify the theoretical findings.
