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

arXiv:2601.02561v1 Announce Type: new
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.