Graphics Processing Unit (GPU) computing is becoming an alternate computing platform for numerical simulations. However, it is not clear which numerical scheme will provide the highest computational efficiency for different types of problems. To this end, numerical accuracies and computational work of several numerical methods are compared using a GPU computing implementation. The Correction Procedure via Reconstruction (CPR), Discontinuous Galerkin (DG), Nodal Discontinuous Galerkin (NDG), Spectral Difference (SD), and Finite Volume (FV) methods are investigated using various reconstruction orders. Both smooth and discontinuous cases are considered for two-dimensional simulations. For discontinuous problems, MUSCL schemes are employed with FV, while CPR, DG, NDG, and SD use slope limiting. The computation time to reach a set error criteria and total time to complete solutions are compared across the methods. It is shown that while FV methods can produce solutions with low computational times, they produce larger errors than high-order methods for smooth problems at the same order of accuracy. For discontinuous problems, the methods show good agreement with one another in terms of solution profiles, and the total computational times between FV, CPR, and SD are comparable.