  This paper presents the development of an Adaptive Algebraic Multiscale
Solver for Compressible flow (C-AMS) in heterogeneous porous media. Similar to
the recently developed AMS for incompressible (linear) flows [Wang et al., JCP,
2014], C-AMS operates by defining primal and dual-coarse blocks on top of the
fine-scale grid. These coarse grids facilitate the construction of a
conservative (finite volume) coarse-scale system and the computation of local
basis functions, respectively. However, unlike the incompressible (elliptic)
case, the choice of equations to solve for basis functions in compressible
problems is not trivial. Therefore, several basis function formulations
(incompressible and compressible, with and without accumulation) are considered
in order to construct an efficient multiscale prolongation operator. As for the
restriction operator, C-AMS allows for both multiscale finite volume (MSFV) and
finite element (MSFE) methods. Finally, in order to resolve high-frequency
errors, fine-scale (pre- and post-) smoother stages are employed. In order to
reduce computational expense, the C-AMS operators (prolongation, restriction,
and smoothers) are updated adaptively. In addition to this, the linear system
in the Newton-Raphson loop is infrequently updated. Systematic numerical
experiments are performed to determine the effect of the various options,
outlined above, on the C-AMS convergence behaviour. An efficient C-AMS strategy
for heterogeneous 3D compressible problems is developed based on overall CPU
times. Finally, C-AMS is compared against an industrial-grade Algebraic
MultiGrid (AMG) solver. Results of this comparison illustrate that the C-AMS is
quite efficient as a nonlinear solver, even when iterated to machine accuracy.
