Modelling detailed localized shallow-water hydrodynamics using a new meshless subdomain solver within TELEMAC-2D

High-resolution modelling of complex, non-linear shallow water flows, such as those interacting with intricate coastal topographies or infrastructure, can require exceptionally fine spatial discretization. Resolving these highly localized dynamics across large, regional-scale domains using traditional mesh-based methods is often computationally prohibitive. To bridge this gap, we present a novel meshless Non-Linear Shallow Water (NLSW) solver implemented as a one-way coupled subdomain within a larger, regional TELEMAC-2D model. The core meshless solver utilizes a hybrid Godunov-Collocation numerical scheme with a Weighted Least Squares (WLS) quadratic basis to efficiently solve the NLSW equations. Spatial stabilization is achieved through a localized artificial pressure term and an Harten-Lax-van Leer (HLL) Riemann solver, which naturally and robustly handles wetting and drying transitions over complex terrain. The solver has been tested on several hydrodynamic test cases - from small flume experiments to the Malpasset dam break - spanning very different scales and conditions, all without any need for parameter tuning.
The integration with TELEMAC-2D is designed for user flexibility and computational efficiency. The subdomain interface is defined directly through the standard TELEMAC-2D steering (.cas) file, which passes a user-defined boundary polygon and high-resolution Digital Elevation Model (DEM) data to the solver. Within this designated polygon, a scattered 2D nodal distribution is rapidly generated. This includes support for advanced pre-processing features, such as image-based mesh injection, allowing users to tailor node density precisely to specific topographic features.
During the simulation, the overarching TELEMAC-2D domain provides hydrodynamic boundary conditions to the meshless subdomain. The meshless model is called by TELEMAC-2D at every global time-step. However, in order to resolve highly transient local physics, the solver operates on a dynamically computed, sub-time step. This satisfies stringent local CFL conditions without imposing restrictive time steps on the global regional model.
While the primary solver is optimized for the hydrostatic NLSW equations, the numerical architecture is designed to be fully extensible. For scenarios requiring the resolution of dispersive wave phenomena, the solver seamlessly accommodates a non-hydrostatic predictor-corrector sequence to capture Boussinesq-type effects.
Ultimately, this nested meshless approach allows for the highly accurate, stable resolution of nonlinear hydrodynamics in targeted areas of interest without the computational overhead of complex global mesh refinement.