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

Research output: Contribution to journalResearch articleContributedpeer-review



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 geometrically Intrinsic Shallow Water Equations (ISWE) are characterized by non-autonomous fluxes. Handling of non-autonomous fluxes is an open question for schemes based on Riemann 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 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.


Original languageEnglish
Article number127776
JournalApplied mathematics and computation
Publication statusPublished - Apr 2023

External IDs

Mendeley 6c3241f6-6a7e-3c4b-8f8b-89e8cbc4c034
Scopus 85145657148
WOS 000908984800001



  • Balance laws on surface, Intrinsic lagrangian-Eulerian scheme, No-flow surfaces, Non-autonomous fluxes, Shallow water equations, Spatially variable topography