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120 Fine Grid 33 grid AIAA Paper Appendix area ratio Brandt coarse grid Computational Fluid Dynamics Conference on Numerical Convergence history Convergence rates defined difference equations difference operators dimensional nozzle flow discontinuous first derivative eigenvalues Euler equations finest grid Finite Difference five point grid Fluid Dynamics flux vector split full approximation Grid Iterations Fig grid M-l ICASE implicit scheme indicated in figure iterative convergence Jacobian matrices l2 percent thick linear interpolation Mach number matrix Methods in Fluid minus flux vector Mth grid Multi-Level Adaptive multigrid method multigrid procedure multigrid relaxation multigrid scheme multigrid sweep NASA nonlifting biconvex airfoil Numerical Methods Numerical Solution obtained partial differential equations Potential Flow Problems rate of convergence relaxation algorithms Relaxation Methods relaxation parameters shown in figure sonic line split forms Steger Subcritical Flow surrogate equation approach Transonic Potential variable vector split schemes