Residual Distribution for Shallow Water Flows

The residual distribution framework was developed as an alternative to the finite volume approach for approximating hyperbolic systems of conservation laws which would allow a natural representation of genuinely multidimensional flow features. The resulting algorithms are closely related to conforming finite elements, but their structure makes it far simpler to construct nonlinear approximation schemes, and therefore to avoid unphysical oscillations in the numerical solution. They have been successfully applied to a wide range of nonlinear systems of equations, producing accurate simulations of both steady and, more recently, time-dependent flows (see [1, 3] for fairly recent reviews of the development of these algorithms). When designed carefully, residual distribution schemes have the following very useful properties. • They can be simultaneously second order accurate (in space and time) and free of unphysical oscillations, even in the presence of turning points in the solution. Even higher order accuracy can also be achieved, but a form of limiting has to be applied, which reverts to the second order scheme, where unwanted oscillations would otherwise appear. This differs from finite volume schemes, which almost invariably drop to first order accuracy when limiters are applied. • The CRD (Conservative Residual Distribution) formulation [2] provides a very natural way to approximate balance terms in a manner which automatically retains equilbria inherent in the underlying system. • It is possible to construct residual distribution schemes which allow for a discontinuous representation of the dependent variables [4]. In particular, the inclusion of discontinuities in time allows for the development of schemes which are unconditionally positive [5], i.e. they are free of unphysical oscillations whatever size of time-step is taken. This presentation will discuss the application of residual distribution to steady and time-dependent shallow water flows. It will include an overview of recent work, such as that of Ricchiuto and Bollermann [6], and progress on the development of a second order accurate, unconditionally positive, well-balanced scheme for the twodimensional shallow water equations with variable bed topography.