  Accurate high-speed flow simulations of practical interest require numerical
methods with high-resolution properties. In this paper, we present an extension
and demonstration of the high-accuracy Gradient-based reconstruction and
$\alpha$-damping schemes introduced by Chamarthi (2022) [1] for simulating
high-speed flows in generalized curvilinear and dynamic mesh domains with the
freestream preservation property. In the first part of this paper, the
algorithms are detailed within the generalized curvilinear coordinate
framework, with a focus on demonstration through stationary and dynamic mesh
test cases. It has been shown both theoretically and through the use of test
cases that the conservative metrics, including their interpolation to cell
interfaces, must be numerically computed using a central scheme that is
consistent with the inviscid flux algorithm to achieve the freestream
preservation property. The second part of the paper illustrates the efficacy of
the algorithm in simulating supersonic jet screech by displaying its capability
to capture the screech tones and accurately characterize the unsteady lateral
flapping mode of a Mach 1.35 under-expanded supersonic jet, in contrast to the
WENO-Z scheme which fails to do so at the same grid resolution. In the final
part of the paper, the parallelizability of the schemes on GPU architectures is
demonstrated and performance metrics are evaluated. A significant speedup of
over $200 \times$ (compared to a single core CPU) and a reduction in simulation
completion time to 34.5 hours per simulation were achieved for the supersonic
jet noise case at a grid resolution of 13 million cells.
