A Schrödinger-Based Dispersive Regularization Approach for Numerical Simulation of One-Dimensional Shallow Water Equations

TL;DR

This work tackles numerical simulation of the one-dimensional shallow water equations (SWE) in regimes involving wetting-drying and vacuum states. It introduces a dispersive regularization via an energetic variational framework, leading to a defocusing cubic nonlinear Schrödinger equation with an external potential $gb(x)$, solved for $\psi$ and postprocessed to obtain $h=|\psi|^2$ and $q=\varepsilon\mathrm{Im}(\overline{\psi}\,\psi_x)$. The authors develop a computational pipeline using a spectral-element spatial discretization and Strang splitting in time, demonstrating $O(\varepsilon)$ accuracy in subcritical, shock-free scenarios including moving wetting-drying fronts and vacuum formation, while revealing that shocks in the original SWE manifest as dispersive shocks in the Schrödinger formulation. The approach yields a robust alternative for SWE with dry interfaces and provides a pathway for higher-dimensional extensions and rigorous error analysis in future work.

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 Schrödinger equation with a drift term induced by bottom topography. Instead of solving the shallow water equations directly, we solve the associated Schrödinger 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 Schrödinger 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 Schrödinger-based formulation offers a robust and promising alternative framework for the numerical simulation of shallow water flows with dry states.

Figures (5)

• Figure 1: Numerical solution of the dam-break problem with a dry bed described in Subsection \ref{['dry_bed']}. The top row shows the real and imaginary parts of the Schrödinger wave function $\psi$. The bottom row displays the reconstructed hydrodynamic variables, namely the water height $h=|\psi|^2$ and the discharge $q=\varepsilon\,\mathrm{Im}(\overline{\psi}\,\psi_x)$. The numerical solutions are compared with the corresponding dispersionless shallow water Riemann solution.
• Figure 2: Numerical solution of the dam-break problem with a wet bed described in Subsection \ref{['wet_bed']}. The top row shows the real and imaginary parts of the Schrödinger wave function $\psi$. The bottom row displays the reconstructed hydrodynamic variables, namely the water height $h=|\psi|^2$ and the discharge $q=\varepsilon\,\mathrm{Im}(\overline{\psi}\,\psi_x)$. The numerical solutions are compared with the corresponding dispersionless shallow water Riemann solution.
• Figure 3: Numerical solution of the vacuum generation problem described in Subsection \ref{['vac']}. The top row shows the real and imaginary parts of the Schrödinger wave function $\psi$. The bottom row displays the reconstructed hydrodynamic variables, namely the water height $h=|\psi|^2$ and the discharge $q=\varepsilon\,\mathrm{Im}(\overline{\psi}\,\psi_x)$. The numerical solutions are compared with the corresponding dispersionless shallow water Riemann solution.
• Figure 4: Numerical solution of the oscillating lake problem described in Subsection \ref{['bowl']}. The top row shows the real and imaginary parts of the Schrödinger wave function $\psi$. The bottom row displays the reconstructed free-surface elevation $\eta(x)=h(x)+b(x)$, together with the bottom topography $b(x)$. The left, middle, and right columns correspond to times $t=2.0$, $t=3.0$, and $t=4.0$, respectively.
• Figure 5: Numerical solution at time $t= 1.0$ of the well-balanced problems described in Subsection \ref{['wb']}. The top row shows the real and imaginary parts of the Schrödinger wave function $\psi$. The middle row displays the reconstructed free-surface elevation $\eta(x)=h(x)+b(x)$, where $h=|\psi|^2$, together with the bottom topography $b(x)$. The bottom row shows the reconstructed free-surface elevation $\eta(x)=h(x)+b(x)$ without the bottom topography, in order to highlight the small deviations of the numerical solution from the exact (dispersionless) steady state. The left column corresponds to the fully wet case with $b_{\max}=0.9$, while the right column corresponds to the partially dry case with $b_{\max}=1.1$.