A geometrically intrinsic lagrangian-Eulerian scheme for 2D shallow water equations with variable topography and discontinuous data

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. This work was dictated by the fact that geo-metrically Intrinsic Shallow Water Equations (ISWE) are characterized by non-autonomous fluxes. Handling of non-autonomous fluxes is an open question for schemes based on Rie-mann solvers (exact or approximate). Using a local curvilinear reference system anchored on the bottom surface, we develop an effective first-order and high-resolution space-time discretization of the no-flow surfaces and solve a Lagrangian initial value problem that de-scribes the evolution of the balance laws governing the geometrically intrinsic shallow wa-ter equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme main-tains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometri-cally intrinsic shallow water equation on variable topographies. We provide a representa-tive set of numerical examples to illustrate the accuracy and robustness of the proposed Lagrangian-Eulerian formulation for two-dimensional surfaces with general curvatures and discontinuous initial conditions.(c) 2022 Elsevier Inc. All rights reserved.