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New versions of implicit algorithms are proposed for the efficient solution of the Euler equations of compressible flow. The methods are based on a preconditioned, Lower-Upper (LU) implementation of a nonlinear, Symmetric Gauss-Seidel (SGS) algorithm for use as a smoothing algorithm in a multigrid method. The methods have been implemented for flows in quasi-one-dimensional ducts and for two dimensional flows past airfoils on boundaryconforming "O"-type grids for a variety of Symmetric Limited Positive (SLIP) spatial approximations, including the scalar dissipation and Convective Upwind Split Pressure (CUSP) schemes. The method is demonstrated to be significantly faster than earlier explicit or implicit methods for this class of problems, allowing solution of these problems to the level of truncation error in three to five multigrid cycles.

How many steps are required to solve the Euler equations of steady, compressible flow - In search of a fast solution algorithm