The aim of the project is to develop new algorithms for solving the Navier-Stokes equations governing viscous compressible flow for upwind discretization of the advection terms based on multi-dimensional wave decomposition and development of multigrid techniques using solution adaptive nested grids.
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.
The major tasks have been summarised as follows:
Implementation and further investigation of genuinely multi-dimensional upwind schemes for the convective terms, both cell centred finite volume and cell vertex fluctuation splitting. Upwinding based on (a) optimal decoupling of the Euler terms, and (b) on Roe's multi-dimensional wave decomposition. The multi-dimensional approach aims to cure some severe weakness in the theory behind state of the art methods which are based on 1-dimensional Riemann solvers.
Development of a multigrid Navier-Stokes code for embedded quadrilateral refining in a tree structure with datastructure suitable for cell-vertex and cell-centred unknowns.
Development of improved smoothing methods for use in the multigrid solver. Investigation of implicit relaxation smoothers. Investigation of new explicit optimal smoothing multi-stage schemes with coefficients and timescale optimized for each characteristic in the multi-dimensional decomposition.
Study of the criterion for solution adaptive grid refining. Development of criteria based on local truncation error estimates using multigrid-extrapolation.
All essential goals of the project can be reached by investigating 2-dimensional laminar flows only. It has been carefully taken into account that extensions to 3 space dimensions require no major new innovations.
Funding SchemeCSC - Cost-sharing contracts
1090 GB Amsterdam