The convergence and accuracy of gradient values on high aspect ratio grids remain problems in CFD. One of the methods solving these problems is to use a hyperbolic system. In this study, we investigated time evolution methods for hyperbolic systems and compare the hyperbolic method with a traditional method. We solve the following test cases: one and two dimensional advection-diffusion problems, Navier-Stokes problems such as laminar flow on a flat plate and laminar flow around a cylinder. We confirmed that the convergence in hyperbolic systems was much more rapid and the accuracy of gradient values was higher than that of traditional system. The hyperbolic system takes almost the same time or shorter time than traditional system on same grids. In the case of Navier-Stokes problems such as high Reynolds number boundary flow, on grids achieving the same accuracy, it takes less time in hyperbolic systems than in traditional systems. One of the major findings is that using approximate Jacobian gives the same order accuracy as using exact Jacobian and reduces calculation time remarkably in hyperbolic system. Calculation time was 19% shorter in 1D advection-diffusion problem, 9% in 2D advection-diffusion problem, and more than 74% in Navier-Stokes systems.