Finite volume numerical methods have been widely studied, implemented and parallelized on multiprocessor systems or on clusters. Modern graphics processing units (GPU) provide architectures and new programing models that enable to harness their large processing power and to design computational fluid dynamics simulations at both high performance and low cost. We report on solving the 2D compressible Euler equations on modern Graphics Processing Units (GPU) with high-resolution methods, i.e. able to handle complex situations involving shocks and discontinuities. We implement two different second order numerical schemes, a Godunov-based scheme with quasi-exact Riemann solver and a fully discrete second-order central scheme as originally proposed by Kurganov and Tadmor. Performance measurements show that these two numerical schemes can achieves x30 to x70 speed-up on recent GPU hardware compared to a mono-thread CPU reference implementation. These first results provide very promising perpectives for designing a GPU-based software framework for applications in computational astrophysics by further integrating MHD codes and N-body simulations.