An incremental block-line-Gauss-Seidel method for the Navier-Stokes equations
NASA, Вашингтон, West Virginia, United States AIAA Journal
(Impact Factor: 1.21).
06/1986; 24(5). DOI: 10.2514/3.9344
A block-line-Gauss-Seidel (LGS) method is developed for solving the incompressible and compressible Navier-Stokes equations in two dimensions. The method requires only one block-tridiagonal solution process per iteration and is consequently faster per step than the linearized block-alternating-direction-implicit (ADI) methods. Results are presented for both incompressible and compressible flows: in all cases the proposed block-LGS method is more efficient than the block-ADI methods. Furthermore, for high-Reynolds-number weakly separated incompressible flow in a channel, which proved to be an impossible task for a block-ADI method, solutions have been obtained very efficiently using the new scheme.
Available from: Michele Napolitano
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ABSTRACT: This paper provides a novel incremental multigrid strategy for the equations of fluid dynamics. The (time dependent) governing equations are discretized in time by means of a two level implicit Euler scheme and linearized using Taylor series and the incremental (delta) form of Beam and Warming. The coefficients and the right hand side of the resulting linear systems are evaluated always at the finest grid level, whereas the (delta) unknowns are computed (approximately, by a single relaxation sweep) on a sequence of coarser meshes. At every grid level the computed deltas are interpolated up to the finest-grid level and used to update the solution, as well as the coefficients and the right hand side of the linear systems. This process is repeated, sweeping all grid levels successively, until a satisfactory convergence criterion is met. The validity of the proposed approach is demonstrated by solving a simple linear problem and the vorticity-stream function Navier-Stokes equations, using line relaxation methods as smoothers, and the lambda-formulation Euler equations, in conjunction with a simple explicit smoother. In all cases, the proposed multigrid strategy provides a considerable efficiency gain over the corresponding single-grid methods.
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ABSTRACT: This paper provides a simple, efficient and robust numerical technique for solving 2-D incompressible steady viscous flows at moderate-to-high Reynolds numbers. The proposed approach employs an incremental multigrid method and an extrapolation procedure based on minimum residual concepts to accelerate the convergence rate of a robust block-line-Gauss-Seidel solver for the vorticity-stream function Navier-Stokes equations.Results are presented for the driven cavity flow problem using uniform and nonuniform grids and for the flow past a backward facing step in a channel. For this second problem, mesh refinement and Richardson extrapolation are used to obtain useful benchmark solutions in the full range of Reynolds numbers at which steady laminar flow is established.
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