Abstract: Runge-Kutta and GMRES methods are used for solving the 3-Dtime-dependent Euler equations in an Arbitrary Lagrangian-Eulerian(ALE)framework. The algorithm is based on a cell centered, finite-volume approach,second-order accurate in space. Hanel method is used to calculate the fluxof the control face. Flows around a pitching NACA0012 airfoil and a moving ballare simulated. The numerical results are satisfactory. GMRES has theadvantages of taking much less memory and less computation time.