A semi-implicit exactly fully well-balanced relaxation scheme for the Shallow Water Linearized Moment Equations
C. Caballero-Cárdenas, I. Gómez-Bueno, A. del Grosso, J. Koellermeier, T. Morales de Luna
TL;DR
This work addresses accurate, efficient simulation of shallow flows with vertical velocity variation by employing the Shallow Water Linearized Moment Equations (SWLME). It introduces a semi-implicit, second-order, exactly fully well-balanced relaxation scheme that splits acoustic and material phenomena, enabling implicit treatment of pressure and explicit transport to relax CFL constraints in subcritical flows. The method uses a fully well-balanced reconstruction operator to preserve steady states with nonconstant vertical velocity profiles and leverages Strang splitting for second-order accuracy. Numerical experiments demonstrate machine-precision well-balancedness, substantial speedups compared to explicit schemes, and robust performance across lake-at-rest, subcritical, perturbation, and dam-break scenarios, validating both accuracy and efficiency. The approach offers a practical, high-fidelity tool for subcritical shallow-water applications where vertical velocity structure is important.
Abstract
When dealing with shallow water simulations, the velocity profile is often assumed to be constant along the vertical axis. However, since in many applications this is not the case, modeling errors can be significant. Hence, in this work, we deal with the Shallow Water Linearized Moment Equations (SWLME), in which the velocity is no longer constant in the vertical direction, where a polynomial expansion around the mean value is considered. The linearized version implies neglecting the non-linear terms of the basis coefficients in the higher order equations. As a result, the model is always hyperbolic and the stationary solutions can be more easily computed. Then, our objective is to propose an efficient, accurate and robust numerical scheme for the SWLME model, specially adapted for low Froude number situations. Hence, we describe a semi-implicit second order exactly fully well-balanced method. More specifically, a splitting is performed to separate acoustic and material phenomena. The acoustic waves are treated in an implicit manner to gain in efficiency when dealing with subsonic flow regimes, whereas the second order of accuracy is achieved thanks to a polynomial reconstruction and Strang-splitting method. We also exploit a reconstruction operator to achieve the fully well-balanced character of the method. Extensive numerical tests demonstrate the well-balanced properties and large speed-up compared to traditional methods.