Turbulence Simulation on Sparse Grids using the Combination

In this paper, we study the parallel numerical solution of the Navier-Stokes equations with the sparse grid combination method. This algorithmic concept is based on the independent solution of many problems with reduced size and their linear combination. We describe the algorithm for three-dimensional problems and we report on its application to turbulence simulation. Furthermore, statistical results on a pipe ow for Reynolds number Re cl = 6950 are presented and compared to results obtained from other numerical simulations and physical experiments. 1. Summary Sparse grid techniques are very promising for the solution of linear PDEs. There, for three-dimensional problems, only O(h ?1 m (log(h ?1 m)) 2) grid points are needed instead of O(h ?3 m) grid points like in the conventional full grid case. Here, h m = 2 ?m denotes the mesh size in the unit cube. However, the accuracy of the sparse grid solution is of the order O(h 2 m (log(h ?1 m)) 2) (with respect to the L 2 ? and L 1 ?norm) provided that the solution is suuciently smooth (i.e. j 6 u(x;y;z) x 2 y 2 z 2 jj 1). This is only slightly worse than the order O(h 2 m) obtained for the usual full grid solution. For further details on sparse grids, see 4], 9]. For the solution of problems arising from the sparse grid discretization approach, two diierent methods have been developed in the last years. First, multilevel-type solvers for the system that results from hierarchical basis like nite element//nite volume or nite diierence discretization (FE, FV, FD) and second, the so-called combination method. There, the solution is obtained on a sparse grid by a certain linear combination of discrete solutions on diierent meshes. The multilevel-type solvers need hierarchical data structures. Thus, specially designed solvers are necessary (see 3]). In the case of the combination method, however, only simple data structures are needed. This method merely handles three-dimensional arrays. Furthermore, it is possible to use any PDE solver as a "black-box" solution method within the combination approach to get a solution on the diierent meshes involved.