In this study, we focus on the simulation of transonic flows in the context of turbomachinery applications, employing a high-order Discontinuous Galerkin Flux Reconstruction (DG-FR) methodology. Conducting high-order simulations within the context of transonic turbines is challenging due to the presence of extreme unsteadiness in the flow accompanied by fluid compressibility effects, such as transonic vortex shedding and unsteady shock wave-boundary layer interactions. These phenomena significantly influence the aerodynamic performance of the airfoils playing a major role also in the determination of the losses in the flow. The primary objective of this work is to evaluate the impact of the formulated non-reflective boundary conditions and of a recently proposed shock capturing scheme on crucial aerodynamic predictions, including the isentropic Mach number and the skin friction distribution. The non-reflective boundary condition is specifically designed to leverage modern Graphical Processing Unit (GPU) architectures, aiming to minimize the necessity for blocking communication. The outcomes of this new outlet boundary condition will be thoroughly analyzed in terms of both computational cost and its influence on the resolved flowfield. The developed physics-based shock-capturing scheme is particularly attractive due to its reduced computational cost and ease of implementation and usage. The focus will be on assessing its performance in the context of highly unsteady flows, such as the current low-pressure turbine cascade. The effectiveness of the proposed computational setup will be rigorously examined to ensure its suitability for simulating complex transonic flows in turbomachinery applications.