  This work studies the parallelization and empirical convergence of two finite
difference acoustic wave propagation methods on 2-D rectangular grids, that use
the same alternating direction implicit (ADI) time integration. This ADI
integration is based on a second-order implicit Crank-Nicolson temporal
discretization that is factored out by a Peaceman-Rachford decomposition of the
time and space equation terms. In space, these methods highly diverge and apply
different fourth-order accurate differentiation techniques. The first method
uses compact finite differences (CFD) on nodal meshes that requires solving
tridiagonal linear systems along each grid line, while the second one employs
staggered-grid mimetic finite differences (MFD). For each method, we implement
three parallel versions: (i) a multithreaded code in Octave, (ii) a C++ code
that exploits OpenMP loop parallelization, and (iii) a CUDA kernel for a NVIDIA
GTX 960 Maxwell card. In these implementations, the main source of parallelism
is the simultaneous ADI updating of each wave field matrix, either column-wise
or row-wise, according to the differentiation direction. In our numerical
applications, the highest performances are displayed by the CFD and MFD CUDA
codes that achieve speedups of 7.21x and 15.81x, respectively, relative to
their C++ sequential counterparts with optimal compilation flags. Our test
cases also allow to assess the numerical convergence and accuracy of both
methods. In a problem with exact harmonic solution, both methods exhibit
convergence rates close to 4 and the MDF accuracy is practically higher.
Alternatively, both convergences decay to second order on smooth problems with
severe gradients at boundaries, and the MDF rates degrade in highly-resolved
grids leading to larger inaccuracies. This transition of empirical convergences
agrees with the nominal truncation errors in space and time.
