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

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.