Three-dimensional staggered grids for systems of hyperbolic conservation laws
- Dreidimensionale versetzte Gitter fuer Systeme hyperbolischer Erhaltungssaetze
Rosenbaum, Wolfram; Noelle, Sebastian (Thesis advisor)
Aachen : Publikationsserver der RWTH Aachen University (2007)
Dissertation / PhD Thesis
Aachen, Techn. Hochsch., Diss., 2007
In 1990 Nessyahu and Tadmor proposed an impressively simple Riemann-solver free Finite Volume staggered grid scheme for the approximate solution of systems of conservation laws in one spatial dimension. The scheme itself is a natural extension of the first order Lax-Friedrichs scheme to second order accuracy. The better approximation order is achieved by two additional ingredients: a) data reconstruction to represent the numerical solution as a cellwise linear function instead of cellwise constant and b) approximating integrals by appropriate second order accurate quadrature. Inspired by this fundamental work, many authors contributed to the extension of the Nessyahu-Tadmor scheme on two and even three spatial dimensions. Whereas in one spatial dimension the structure of the computational grid is simple and has only minor impact on the formulation of the finite volume scheme, the geometrical description of the grid becomes more important in higher spatial dimensions, especially when grid cells may differ in size and shape. In  we addressed the problem of constructing staggered grids on adaptive cartesian meshes in 2D. Taking advantage of the cartesian grid structure one defines a (small) set of local geometrical patterns which pieced together describe the geometry of the staggered grid. In the present work we generalized and extended our 2D pattern approach to three-dimensional adaptive cartesian grids. Local patterns are defined by a Voronoi decomposition of cartesian grid cells respecting their direct neighbours. On adaptive meshes local grid refinement and thus level transition between adjacent grid cells results in a variety of local patterns. Applying Polya's theory of counting, the number of combinatorially different patterns could be determined to equal only 227. Though not mandatory, the a priori knowledge of all possible patterns proved to be extremly helpful for the implementation of the numerical solver. Here, we adopted the Nessyahu-Tadmor scheme to three-dimensional staggered grids. The staggered grid constructed by piecing together local patterns in general lacks the cartesian structure. Therefore the data reconstruction applies a least squares technique primiraly proposed for unstructured grids. Quadrature rules for second order accurate integration have been adjusted to cope with general polyhedral cell shapes. With regard to real CFD-applications the grid geometry and the staggered grid scheme have been modified to incorporate complex obstacles. The resulting finite volume scheme has been applied to advection problems (scalar conservation law) and problems in fluid dynamics (stated in terms of the Euler equations as a system of conservation laws).  W. Rosenbaum, Ein zweidimensionales adaptives staggered-grid Verfahren zur Loesung von Systemen hyperbolischer Differentialgleichungen. Bonn University, 1999.