Very high order well-balanced schemes for non-prismatic one-dimensional channels with arbitrary shape

Accurate modelling of shallow water flows in canals for realistic scenarios cannot neglect the variations of the size and shape of the cross-section along the canal. In typical situations, especially in view of applications to flows in networks of canals, a 2D model would be too costly, while the standard 1D Saint-Venant model for a rectangular channel with constant breadth would be too coarse. In this paper we derive efficient, high order accurate and robust numerical schemes for a 1.5D model, in which the canal can have an arbitrary cross-section: the shape can vary along the channel and is described by a depth-dependent breadth function σ(x,z), where x is the coordinate along the channel and z represents the vertical direction. Contrary to all previous schemes for this model, we reformulate the equations in a way that avoids the appearance of the moments of σ and of ∂σ/∂x in the source terms. Our numerical schemes are based on the path-conservative approach for dealing with non-conservative products, are well-balanced on the lake at rest solution and can treat wet-try transitions. They can be implemented with any order of accuracy. Schemes up to third order are explicitly constructed and tested, thanks to the CWENO reconstruction technique. Through a large set of numerical tests we show the performance of the new schemes and compare the results obtained with different orders of accuracy.