  We have developed a fully sparse, compact-scheme based biglobal stability
analysis numerical solver applied, for the scope of the current paper, to the
investigation of the effects of impedance boundary conditions (IBCs) on the
structure of a fully developed compressible turbulent channel flow. A
sixth-order compact finite difference scheme is used to discretize the
linearized Navier-Stokes equations leading to a Generalized Eigenvalue Problem
(GEVP). Sparsity is retained by explicitly introducing derivatives of the
perturbation as additional unknowns, increasing the overall problem size
(number of columns $\times$ number of rows) while significantly reducing the
number of non-zeros and the computational cost with respect to traditional
implementations yielding otherwise dense matrix blocks. The resulting GEVP is
coded in Python and solved employing an Message Passing Interface (MPI)
parallelized PETSc-based sparse eigenvalue solver adopting a modified Arnoldi
algorithm. Base flow is taken from impermeable isothermal-wall turbulent
channel flow simulations at bulk Reynolds number, $Re_b= 6900$ and Mach number,
$M_b = 0.85$. The eigenvalue spectrum calculated in the biglobal stability
analysis shows distinct groups of modes associated with a discrete set of
streamwise wave numbers accommodated by computational domain. An iterative
strategy for the imposition of the complex IBCs, which are a nonlinear function
of the real-valued (Fourier) frequency in the GEVP, has been devised. The
adopted IBC specifically represents an array of sub-surface-mounted Helmholtz
cavities with resonant frequency, $f_{res}$, covered by a porous sheet with
permeability inversely proportional to the impedance resistance $R$. The
tunable resonant frequency has been shown to be an attractor for the
instability, yielding a single unstable mode at that frequency.
