A Schr\"odinger-Based Dispersive Regularization Approach for Numerical Simulation of One-Dimensional Shallow Water Equations

TL;DR

This paper introduces a dispersive regularization method for simulating 1D shallow water equations by transforming them into a Schr"odinger equation, enabling efficient and robust numerical solutions especially with dry states.

Contribution

The authors develop a novel Schr"odinger-based regularization framework that simplifies the numerical simulation of shallow water equations, particularly handling dry states effectively.

Findings

Achieves $O(\varepsilon)$ approximation in subcritical regimes.

Maintains convergence even with moving wetting-drying interfaces.

Offers a robust alternative to traditional shallow water solvers.

Abstract

We propose a novel dispersive regularization framework for the numerical simulation of the one-dimensional shallow water equations (SWE). The classical hyperbolic system is regularized by a third-order dispersive term in the momentum equation, which renders the system equivalent, via the Madelung transform, to a defocusing cubic nonlinear Schr\"odinger equation with a drift term induced by bottom topography. Instead of solving the shallow water equations directly, we solve the associated Schr\"odinger equation and recover the hydrodynamic variables through a simple postprocessing procedure. This approach transforms the original nonlinear hyperbolic system into a semilinear complex-valued equation, which can be efficiently approximated using a Strang time-splitting method combined with a spectral element discretization in space. Numerical experiments demonstrate that, in subcritical regimes without shock formation, the Schr\"odinger regularization provides an $O(\varepsilon)$ approximation to the classical shallow water solution, where $\varepsilon$ denotes the regularization parameter. Importantly, we observe that this convergence behavior persists even in the presence of moving wetting--drying interfaces, where vacuum states emerge and standard shallow water solvers often encounter difficulties. These results suggest that the Schr\"odinger-based formulation offers a robust and promising alternative framework for the numerical simulation of shallow water flows with dry states.