Multi-Scale Approximation of Thin-Layer Flows on Curved Topographies

This paper is devoted to a multi-scale approach for describing the dynamic behaviors of thin-layer flows on complex topographies. Because the topographic surfaces are generally curved, we introduce an appropriate coordinate system for describing the flow behaviors in an e cient way. In the present study, the complex surfaces are supposed to be composed of a finite number of triangle elements. Due to the unequal orientation of the triangular elements, the distinct flux directions add to the complexity of solving the Riemann problems at the boundaries of the triangular elements. Hence, a vertex-centered cell system is introduced for computing the evolution of the physical quantities, where the cell boundaries lie within the triangles and the conventional Riemann solvers can be applied. Consequently, there are two mesh scales: the element scale for the local topographic mapping and the vertex-centered cell scale for the evolution of the physical quantities. The final scheme is completed by employing the HLL-approach for computing the numerical flux at the interfaces. Three numerical examples (the Ritter's solution, the Stoker's solution and the parabolic similarity solution) and one application to a large-scale landslide are conducted to examine the performance of the proposed approach as well as to illustrate its capability in describing the shallow flows on complex topographies.