A new multigrid relaxation scheme is developed for the steady-state solution of the Euler and Navier-Stokes equations. The lower-upper symmetric-Gauss-Seidel method (LUSGS) does not require flux splitting for approximate Newton iteration. The method is vectorizable and unconditionally stable, requiring only scalar diagonal inversions. Application to transonic flow shows that the new method is efficient and robust. The Newton iteration method has been investigated to solve the steady Euler or Navier-Stokes equations. Due to the rapid growth of the operation count with the number of mesh cells, the system was solved indirectly. This paper develops an efficient multigrid relaxation scheme for approximate Newton iteration. The new LU-SGS method permits scalar diagonal inversions, whereas the conventional Gauss-Seidel method requires block matrix inversions. Scalar diagonal inversions offer the potential for order-of-magnitude speedups when solving large systems of partial differential equations. The matrix should be diagonally dominant to ensure convergence of a relaxation method. The new method achieves this without flux splitting. Flux splittings increase computational work per cycle. Unlike the conventional Gauss-Seidel method, the present method in three dimensions does not need additional relaxation or factorization on a plane of sweep. The Navier-Stokes equations represent gas flow in thermodynamic equilibrium. Using a finite-volume method for space discretization allows handling arbitrary geometries and avoids problems with metric singularities. A central-difference scheme can produce flowfield oscillations near shock waves. Adaptive numerical dissipation is used to suppress spurious oscillations and prevent nonphysical overshoots. A total variation diminishing (TVD) scheme can be used for more accurate capturing of oblique shock waves in hypersonic flows. The LU-SGS method for approximate Newton iteration can be derived. The method eliminates the need for block diagonal inversions without using a diagonalization procedure. For the Navier-Stokes equations, F and G are replaced by F - Fv and G - Gv. The LU family of algorithms are vectorizable along i + j = const lines on a vector computer. The underlying idea of a multigrid method is to transfer some of the task of tracking the evolution of the system to a sequence of successively coarser meshes. This reduces computational effort per cycle and helps attain global equilibrium more rapidly. The method was tested on several cases, including inviscid transonic flow, viscous laminar flow, and viscous turbulent flow. The LU-SGS method was combined with a flux-limited TVD scheme and applied to chemically reacting nonequilibrium flows in scramjet combustors.A new multigrid relaxation scheme is developed for the steady-state solution of the Euler and Navier-Stokes equations. The lower-upper symmetric-Gauss-Seidel method (LUSGS) does not require flux splitting for approximate Newton iteration. The method is vectorizable and unconditionally stable, requiring only scalar diagonal inversions. Application to transonic flow shows that the new method is efficient and robust. The Newton iteration method has been investigated to solve the steady Euler or Navier-Stokes equations. Due to the rapid growth of the operation count with the number of mesh cells, the system was solved indirectly. This paper develops an efficient multigrid relaxation scheme for approximate Newton iteration. The new LU-SGS method permits scalar diagonal inversions, whereas the conventional Gauss-Seidel method requires block matrix inversions. Scalar diagonal inversions offer the potential for order-of-magnitude speedups when solving large systems of partial differential equations. The matrix should be diagonally dominant to ensure convergence of a relaxation method. The new method achieves this without flux splitting. Flux splittings increase computational work per cycle. Unlike the conventional Gauss-Seidel method, the present method in three dimensions does not need additional relaxation or factorization on a plane of sweep. The Navier-Stokes equations represent gas flow in thermodynamic equilibrium. Using a finite-volume method for space discretization allows handling arbitrary geometries and avoids problems with metric singularities. A central-difference scheme can produce flowfield oscillations near shock waves. Adaptive numerical dissipation is used to suppress spurious oscillations and prevent nonphysical overshoots. A total variation diminishing (TVD) scheme can be used for more accurate capturing of oblique shock waves in hypersonic flows. The LU-SGS method for approximate Newton iteration can be derived. The method eliminates the need for block diagonal inversions without using a diagonalization procedure. For the Navier-Stokes equations, F and G are replaced by F - Fv and G - Gv. The LU family of algorithms are vectorizable along i + j = const lines on a vector computer. The underlying idea of a multigrid method is to transfer some of the task of tracking the evolution of the system to a sequence of successively coarser meshes. This reduces computational effort per cycle and helps attain global equilibrium more rapidly. The method was tested on several cases, including inviscid transonic flow, viscous laminar flow, and viscous turbulent flow. The LU-SGS method was combined with a flux-limited TVD scheme and applied to chemically reacting nonequilibrium flows in scramjet combustors.