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NS SOLVERS & UPWIND SCHEMES - solution adaptive navier - stokes solvers using multi-dimensional upwin schemes and multigrid acceleration

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The principal goal of this project is to cure some of the weaknesses in existing state-of-the-art methods in algorithm development for the numerical solution of the steady 3-dimensional, compressible Navier-Stokes equations.
This project deals with new trends in algorithm development for the numerical solution of the steady, 3-dimensional, compressible Navier-Stokes equations. Results have been obtained already by developing:
multidimensional upwind discretization techniques for the convection terms;
solution adaptive multigrid techniques for the solution of the discretized equations.
2 specific techniques are still in development in each of these 2 project areas:
multidimensional cell vertex fluctuation splitting techniques;
sparse grid techniques.
First order as well as second order accurate and linear as well as nonlinear fluctuation splitting schemes have been developed and applied. To apply the fluctuation splitting schemes to a hyperbolic system of conservation laws, a wave modelling step has to be added. The design of optimal wave decompositions is currently under way. Promising multidimensional upwind results have been obtained, not only through the cell vertex fluctuation splitting approach, but also through the cell centered finite volume approach. To solve the multidimensional upwind discretized equations by a multigrid method, optimal multistage time stepping schemes were developed. With multigrid techniques it is not always necessary to create a finer grid by halving all cells in all coordinate directions. For some problems, it appears favourable to refine in one direction only: the sparse grid way of refining. Sparse grid techniques can be interpreted asgeneralizations of standard geometric multigrid techniques. A crucial step in implementing sparse grid techniques is the choice and construction of the data structure. The data structure has been constructed and implemented in FORTRAN. Pilot computations for a 3-dimensional Poisson equation are being carried out. The upwind methods considered in this project try to respect as many multidimensional flow features as possible, by extracting a maximum amount of physical information from a minimum amount of numerical data. In terms of computational efficiency per grid point, solution adaptive sparse grid methods appear to be best suited.
Promising research results have been obtained already during the First Phase of the project. This was achieved by developing:

multi-D upwind discretization techniques for the convection terms and
solution adaptive multigrid techniques for the solution of the discretized equations.

Research in these two areas is continued in the preset Second Phase of the project.

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Institut Von Karman de Dynamique des Fluides
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72,Chée de Waterloo
1640 Rhode-Saint-Genése
Belgien

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