High-order finite volume schemes on unstructured grids using moving least-squares reconstruction. Application to shallow water dynamics

This paper introduces the use of moving least-squares (MLS) approximations for the development of high-order finite volume discretizations on unstructured grids. The field variables and their successive derivatives can be accurately reconstructed using this mesh-free technique in a general nodal arrangement. The methodology proposed is used in the construction of two numerical schemes for the shallow water equations on unstructured grids: a centred Lax–Wendroff method with added shock-capturing dissipation, and a Godunov-type upwind scheme, with linear and quadratic reconstructions. This class of mesh-free techniques provides a robust and general approximation framework which represents an interesting alternative to the existing procedures, allowing, in addition, an accurate computation of the viscous fluxes.