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

TL;DR

This paper introduces a geometrically intrinsic Lagrangian-Eulerian numerical scheme for 2D shallow water equations that effectively handles variable topography and discontinuous data, maintaining accuracy and robustness.

Contribution

The paper develops a novel Lagrangian-Eulerian scheme using a local curvilinear reference system for intrinsic shallow water equations with variable bottom geometry.

Findings

Accurately captures shocks without excessive numerical dissipation.

Maintains monotonicity and robustness on complex geometries.

Demonstrates high-resolution and effective handling of discontinuous initial data.

Abstract

We present a Lagrangian-Eulerian scheme to solve the shallow water equations in the case of spatially variable bottom geometry. 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 describes the evolution of the balance laws governing the geometrically intrinsic shallow water equations. The evolved solution set is then projected back to the original surface grid to complete the proposed Lagrangian-Eulerian formulation. The resulting scheme maintains monotonicity and captures shocks without providing excessive numerical dissipation also in the presence of non-autonomous fluxes such as those arising from the geometrically intrinsic shallow water equation on variable topographies. We provide a representative 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.