Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes

Abstract

We consider the numerical solution of the shallow water equations on unstructured grids. We focus on flows over wet areas. The extension to the case of dry bed will be reported elsewhere. The shallow water equations fall into the category of systems of conservation laws which can be symmetrized thanks to the existence of a mathematical entropy coinciding, in this case, with the total energy. Our aim is to show the application of a particular class of conservative residual distribution ( RD ) schemes to the discretization of the shallow water equations and to analyze their discrete accuracy and stability properties. We give a review of conservative RD schemes showing relations between different approaches previously published, and recall L ∞ stability and accuracy criteria characterizing the schemes. In particular, the accuracy of the RD method in presence of source terms is analyzed, and conditions to construct rth order discretizations on irregular triangular grids are proved. It is shown that the RD approach gives a natural way of obtaining high order discretizations which, moreover, preserves exactly the steady lake at rest solution independently on mesh topology, nature of the variation of the bottom and polynomial order of interpolation used for the unknowns. We also consider more general analytical solutions which are less investigated from the numerical view point. On irregular grids, linearity preserving RD schemes yield a truly second order approximation. We also sketch a strategy to achieve discretizations which preserve exactly some of these solutions. Numerical results on steady and time-dependent problems involving smooth and non-smooth variations of the bottom topology show very promising features of the approach.