Multidimensional Upwind Schemes: Application to Hydraulics

Abstract

In the field of the numerical simulation of conservation laws, upwind and TVD techniques have progressively gained acceptance. Originally, they were derived for homogeneous scalar equations or systems of equations in one spatial dimension. Their extension to more than one spatial dimension is not straightforward.

A widespread version is based on a piecewise constant representation of the solution inside cells and the application of a one dimensional Riemann solver across every cell edge. This leads to a finite volume technique based on the integral form of the equations to which the divergence theorem has been applied. In a philosophy different from concentrating on finite volumes and the changes of the variables across the cell sides, it is possible to consider solutions on grids in which the unknowns are associated with the vertices and updates to these nodal values are through the advection of linear wave solutions. This avoids the problems of taking the normal to the cell interfaces as a privileged direction. This second group of methods are based on a piecewise linear continuous representation.

Some years after their adoption for solving problems in gas dynamics, upwind schemes have been successfully used for the solution of the shallow water equations, with similar advantages. We consider the use of these tecniques for 2D shallow water flows and the question of whether they may be of practical use. The basis of the numerical methods is stated and their application to the shallow water system is described. Finally, some numerical results are presented.

Two dimensional wave decomposition and multi-dimensional upwinding seem a promising method of solution for the 2D shallow water equations. Two wave models have been adapted from Gas Dynamics to render the technique suited to hydraulic problems with shocks. Although the procedure is more complicated than present day generalizations of ID upwinding techniques it is cell-based, which makes it competitive versus the edge based finite volume techniques. Both can be applied on a triangular discretization and, by taking advantage of the triangles, they can clearly be applied to arbitrary geometries, a great advantage for hydraulic engineers working on practical problems, and there is a wide variety of possibilities concerning grid movement and refinement.