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We present a Lagrangian-Eulerian scheme to solve the shallow water equationsin the case of spatially variable bottom geometry. Using a local curvilinearreference system anchored on the bottom surface, we develop an effectivefirst-order and high-resolution space-time discretization of the no-flowsurfaces and solve a Lagrangian initial value problem that describes theevolution of the balance laws governing the geometrically intrinsic shallowwater equations. The evolved solution set is then projected back to theoriginal surface grid to complete the proposed Lagrangian-Eulerian formulation.The resulting scheme maintains monotonicity and captures shocks withoutproviding excessive numerical dissipation also in the presence ofnon-autonomous fluxes such as those arising from the geometrically intrinsicshallow water equation on variable topographies. We provide a representativeset of numerical examples to illustrate the accuracy and robustness of theproposed Lagrangian-Eulerian formulation for two-dimensional surfaces withgeneral curvatures and discontinuous initial conditions.

A geometrically intrinsic Lagrangian-Eulerian scheme for 2D Shallow Water Equations with variable topography and discontinuous data