Do Finite Volume Methods Need a Mesh?
In this article, finite volume discretizations of hyperbolic conservation laws are considered, where the usual triangulation is replaced by a partition of unity on the computational domain. In some sense, the finite volumes in this approach are not disjoi
- PDF / 1,743,192 Bytes
- 16 Pages / 439 x 666 pts Page_size
- 16 Downloads / 213 Views
1
Introduction
The finite volume method (FVM) is a standard approach to construct approximate solutions of hyperbolic conservation laws [4,8]. The basic idea is to split the computational domain into small cells - the finite volumes - and to enforce conservation by prescribing fluxes at the cell interfaces: if a certain amount of the conserved quantity leaves cell Gi across a common boundary r ij with cell Gj , it has to reappear in Gj . In this way, the evolution of the conserved quantities can be approximated if the fluxes are suitable approximations of the fluxes given by the conservation laws. At this level, the underlying mesh seems to be very important. However, if one looks at the finite volume method from a more abstract point of view, it appears as a system of ODEs with the following ingredients: a numerical flux function 9 and parameters Vi and J3ij' Here, Vi has the interpretation as volume of cell Gi , lJ3ij l is the surface area of the interface rij , and J3 ij /lJ3ij l is the corresponding normal vector, pointing from cell Gi to cell Gj . Now the question, whether finite volume methods need a mesh, can be reformulated mathematically: under which conditions on Vi and J3 ij does the finite volume method produce reliable approximations to solutions of the conservation law? Is it really necessary that Vi and J3ij are constructed from a mesh, or do they just have to satisfy some algebraic relations? This interesting question naturally arises in connection with the finite volume particle method (FVPM) which has recently been proposed in [5]. At the core of this method is a partition of unity on the computational domain where the partition functions are used as test functions in the weak formulation ofthe conservation law. As a result, a system of ODEs is obtained M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
224
M. Junk
which looks very much like a finite volume method but the parameters Vi and are given by integrals over the partition functions and not as quantities derived from a mesh. Nevertheless, the obtained geometric parameters satisfy all assumptions which are needed in the convergence proof of classical finite volume methods and numerical experiments show that FVPM yields reliable results. Hence, we can say that reasonable geometric parameters Vi, f3ij in finite volume methods can be generated without underlying mesh. In the article, the requirements on the geometric parameters Vi and f3 ij are explained and numerical examples are presented which show the practical relevance of these conditions. Moreover, we show that FVPM can easily be coupled with classical finite volume methods. Apart from FVPM we introduce a quite similar method called PUMESH. Starting from a partition of unity on a d-dimensional domain, we build an associated mesh on a d + 1 dimensional cylinder with the original geometry as cross section. On this grid, a classical finite volume ansatz is used which gives rise to a scheme where the additional dimens
Data Loading...
