Invariant discretization schemes for the shallow-water equations

Invariant discretization schemes are derived for the one- and two-dimensional shallow-water equations with periodic boundary conditions. While originally designed for constructing invariant finite difference schemes, we extend the usage of difference invariants to allow constructing of invariant finite volume methods as well. It is found that the classical invariant schemes converge to the Lagrangian formulation of the shallow-water equations. These schemes require to redistribute the grid points according to the physical fluid velocity, i.e., the mesh cannot remain fixed in the course of the numerical integration. Invariant Eulerian discretization schemes are proposed for the shallow-water equations in computational coordinates. Instead of using the fluid velocity as the grid velocity, an invariant moving mesh generator is invoked in order to determine the location of the grid points at the subsequent time level. The numerical conservation of energy, mass and momentum is evaluated for both the invariant and non-invariant schemes.

💡 Research Summary

The paper develops symmetry‑preserving (invariant) discretization schemes for the one‑ and two‑dimensional shallow‑water equations under periodic boundary conditions. Starting from a Lie‑group analysis of the continuous equations, the authors identify the relevant transformation group (including Galilean invariance and scaling) and construct difference invariants—combinations of grid coordinates, time steps, and field variables that remain unchanged under these transformations. These invariants serve as building blocks for both finite‑difference and finite‑volume formulations.

For the one‑dimensional case, an invariant Lagrangian scheme is derived. Grid points are advected with the fluid velocity, so the mesh moves exactly as the material particles do. The scheme automatically conserves mass because the cell volumes follow the flow, and the difference invariants guarantee that the discrete equations inherit the continuous symmetries. Numerical tests confirm convergence to the Lagrangian form of the shallow‑water system and show excellent energy preservation. However, the moving mesh can become highly distorted, requiring occasional re‑gridding.

In two dimensions the authors extend the invariant construction to multi‑dimensional difference invariants and embed them in a finite‑volume framework. Cell averages are updated while the mesh moves with the flow, and a special reconstruction step ensures that cell areas (or volumes) are preserved, thereby maintaining mass, momentum, and energy conservation at the discrete level. The invariant finite‑volume scheme reproduces the Lagrangian dynamics with reduced numerical diffusion compared to standard Eulerian methods.

To avoid the drawbacks of a purely Lagrangian mesh, the paper introduces invariant Eulerian schemes formulated in computational coordinates. Instead of using the fluid velocity as the mesh velocity, a mesh‑generator equation—derived from the same symmetry requirements—is solved at each time step. This moving‑mesh generator determines the new grid locations while preserving the invariance properties. The resulting Eulerian invariant scheme keeps the mesh regular, limits distortion, and still respects the underlying symmetries. Numerical experiments demonstrate that this approach dramatically reduces energy drift and improves long‑time stability relative to conventional non‑invariant Eulerian schemes.

Finally, the authors perform a systematic comparison between invariant and non‑invariant methods. They evaluate the global errors in mass, momentum, and energy over long integration times. The invariant schemes—both Lagrangian and Eulerian—exhibit markedly smaller conservation errors, with the Eulerian invariant method offering a practical balance between mesh regularity and symmetry preservation. The paper concludes that invariant discretizations, whether implemented as moving‑mesh finite differences or as invariant finite‑volume methods, provide a robust framework for simulating shallow‑water dynamics with superior physical fidelity and numerical stability.