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AF2 scheme AIAA Journal approximate factorization scheme artificial viscosity Ballhaus boundary value calculation Chebyshev Chebyshev-Fourier Problem coarse-grid operator coarse-grid problem coarser level coefficient distribution spectral collocation points compressible flows Computational Fluid Dynamics convergence histories cycle damp denoted Dirichlet boundary conditions efficiency equivalent smoothing rate error components FFT's fine-grid finite difference codes finite difference methods Fourier series full potential equation high-frequency error incomplete LU decomposition interpolation iterative scheme Jameson Kutta condition Lf(Vf lift coefficient lifting airfoil linear problems machine time NACA matrix matrix-multiply Maximum residual MGAFSP multigrid algorithm multigrid scheme Neumann boundary condition points NACA 0012 potential flow problem pressure coefficient distribution pseudospectral method Rectangular Chebyshev-Chebyshev Rectangular Chebyshev-Fourier relaxation scheme SG/AF shown in Figure small perturbation equation spectral discretization spectral methods spectral multigrid methods Streett subcritical flows supercritical flow surface pressure coefficient Table TAIR result technique trailing edge transform transonic flow problems Transonic Potential Flow transonic small perturbation wavenumber Zang