The efficient solution of fluid dynamics problems by the combination technique

Abstract

We study the sparse grid combination technique as an efficient method for the solution of fluid dynamics problems. The combination technique needs only O(h–1n(log(h–1n))d–1) grid points for d‐dimensional problems, instead of O(h–dn) grid points used by the full grid method. Here, hn = 2–n denotes the mesh width of the grids. Furthermore, provided that the solution is sufficiently smooth, the accuracy (with respect to the L2‐ and the L∞‐norm) of the sparse grid combination solution is O(hαn(log(h–1n))d–1), which is only slightly worse than O(hαn) obtained by the full grid solution. Here, α includes the order of the underlying discretization scheme, as well as the influence of singularities. Thus, the combination technique is very economic on both storage requirements and computing time, but achieves almost the same accuracy as the usual full grid solution. Another advantage of the combination technique is that only simple data structures are necessary. Where other sparse grid methods need hierarchical data structures and thus specially designed solvers, the combination method handles merely d‐dimensional arrays. Thus, the implementation of the combination technique can be based on any “black box solver”. However, for reasons of efficiency, an appropriate multigrid solver should be used. Often, fluid dynamics problems have to be solved on rather complex domains. A common approach is to divide the domain into blocks, in order to facilitate the handling of the problem. We show that the combination technique works on such blockstructured grids as well. When dealing with complicated domains, it is often desirable to grade a grid around a singularity. Graded grids are also supported by the combination technique. Finally, we present the first results of numerical experiments for the application of the combination method to CFD problems. There, we consider two‐dimensional laminar flow problems with moderate Reynolds numbers, and discuss the advantages of the combination method.