The paper introduces a new multigrid relaxation scheme, the Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method, for solving the steady-state Euler and Navier-Stokes equations. This method does not require flux splitting for approximate Newton iteration and is vectorizable and unconditionally stable, relying only on scalar diagonal inversions. The LU-SGS method is shown to be efficient and robust through applications to transonic flow, demonstrating a 30% faster convergence rate and a 30% reduction in computational work per cycle compared to the LU implicit scheme. The method is also applied to viscous laminar and turbulent flows, as well as chemically reacting nonequilibrium flows in scramjet combustors. The LU-SGS method's effectiveness is further validated through numerical experiments and comparisons with other methods.The paper introduces a new multigrid relaxation scheme, the Lower-Upper Symmetric-Gauss-Seidel (LU-SGS) method, for solving the steady-state Euler and Navier-Stokes equations. This method does not require flux splitting for approximate Newton iteration and is vectorizable and unconditionally stable, relying only on scalar diagonal inversions. The LU-SGS method is shown to be efficient and robust through applications to transonic flow, demonstrating a 30% faster convergence rate and a 30% reduction in computational work per cycle compared to the LU implicit scheme. The method is also applied to viscous laminar and turbulent flows, as well as chemically reacting nonequilibrium flows in scramjet combustors. The LU-SGS method's effectiveness is further validated through numerical experiments and comparisons with other methods.