The numerical simulation of free surface ¯ows that alternately ¯ood and dry out over complex topography is a formidable task. The model equation set generally used for this purpose is the two-dimensional (2D) shallow water wave model (SWWM). Simpli®ed forms of this system such as the zero inertia model (ZIM) can accommodate speci®c situations like slowly evolving ¯oods over gentle slopes. Classical numerical techniques, such as ®nite dierences (FD) and ®nite elements (FE), have been used for their integration over the last 20±30 years. Most of these schemes experience some kind of instability and usually fail when some particular domain under speci®c ¯ow conditions is treated. The numerical instability generally manifests itself in the form of an unphysical negative depth that subsequently causes a run-time error at the computation of the celerity and/or the friction slope. The origins of this behaviour are diverse and may be generally attributed to:
- The use of a scheme that is inappropriate for such complex ¯ow conditions (mixed regimes). 2. Improper treatment of a friction source term or a large local curvature in topography. 3. Mishandling of a cell that is partially wet/dry.
In this paper, a tentative attempt has been made to gain a better understanding of the genesis of the instabilities, their implications and the limits to the proposed solutions. Frequently, the enforcement of robustness is made at the expense of accuracy. The need for a positive scheme, that is, a scheme that always predicts positive depths when run within the constraints of some practical stability limits, is fundamental. It is shown here how a carefully chosen scheme (in this case, an adaptation of the solver to the SWWM) can preserve positive values of water depth under both explicit and implicit time integration, high velocities and complex topography that may include dry areas. However, the treatment of the source terms: friction, Coriolis and particularly the bathymetry, are also of prime importance and must not be overlooked. Linearization with a combination of switching between explicit±implicit integration can overcome the `stiness' of the friction and Coriolis terms and provide stable numerical integration. The treatment of the bathymetry source term is much more delicate. For cells undergoing a transient wet±dry process, the imposition of zero velocity stabilizes most of the approximations. However, this arti®cial zero velocity condition can be the cause of considerable error, especially when fast moving fronts are involved. Besides these diculties linked with the internal position of the front within a cell versus the limited resolution of a numerical grid, it appears that the second derivative that de®nes whether the bed is locally convex or concave is a key indicator for stability. A convex bottom may lead to unbounded solutions. It appears that this behaviour is not linked to the numerics (numerical scheme) but rather to the mathematical theory of the SWWM. These concerns about stability have taken precedence, until now, over the crucial and related question of accuracy, especially near a moving front, and how these possible inaccuracies at the leading edge may aect the solution at interior points within the domain.
This paper presents an in depth, fully two-dimensional space analysis of the aforementioned problem that has not been addressed before.