Locally Adaptive Non-Hydrostatic Shallow Water Extension for Moving Bottom-Generated Waves

TL;DR

The paper develops a locally adaptive non-hydrostatic extension of the shallow water equations to model moving-bottom-generated waves, leveraging a projection-based predictor-corrector framework where the elliptic correction for the non-hydrostatic pressure is applied only in regions identified by hydrostatic predictor criteria. Validation against analytical solitary-wave solutions and laboratory-moving-bottom experiments demonstrates that the adaptive approach preserves dispersive accuracy while significantly reducing compute time (over 50% in tested scenarios). A systematic study of adaptivity criteria reveals that simple predictors based on the hydrostatic solution, notably the ratio $| ilde{η}|/d$, offer robust performance with favorable accuracy-time trade-offs; enlarging the adaptive region further mitigates interface artifacts. The work points to practical gains for large-scale simulations and lays groundwork for extending to two dimensions and including friction/viscosity effects.

Abstract

We propose a locally adaptive non-hydrostatic model and apply it to wave propagation generated by a moving bottom. This model is based on the non-hydrostatic extension of the shallow water equations (SWE) with a quadratic pressure relation, which is suitable for weakly dispersive waves. The approximation is mathematically equivalent to the Green-Naghdi equations. Applied globally, the extension requires solving an elliptic system of equations in the whole domain at each time step. Therefore, we develop an adaptive model that reduces the application area of the extension and by that the computational time. The elliptic problem is only solved in the area where the dispersive effect might play a crucial role. To define the non-hydrostatic area, we investigate several potential criteria based on the hydrostatic SWE solution. We validate and illustrate how our adaptive model works by first applying it to simulate a simple propagating solitary wave, where exact solutions are known. Following that, we demonstrate the accuracy and efficiency of our approach in more complicated cases involving moving bottom-generated waves, where measured laboratory data serve as reference solutions. The adaptive model yields similar accuracy as the global application of the non-hydrostatic extension while reducing the computational time by more than 50%.

Figures (10)

• Figure 1: Comparison of our numerical simulation results (blue solid line) and its analytical solution (black dashed line) recorded at $t = 0, 10, 20, 30~s$.
• Figure 2: Comparison of the numerical simulations (blue solid line) and measured laboratory data (black dashed line) recorded at $x = 0.61, 1.61, 9.61, 20.61~m$ for the uplift case.
• Figure 3: Comparison of the numerical simulations (blue solid line) and measured laboratory data (black dashed line) recorded at $x = 0.61, 1.61, 9.61, 20.61~m$ for the downdraft case.
• Figure 4: Comparison of the simulation result (blue solid line) with measured laboratory data (black dashed line) for a sliding semi-elliptic plate for $Fr=0.125$ (top), $0.25$ (middle), and $0.375$ (bottom).
• Figure 5: Illustration of adaptive simulation of a propagating solitary wave, with splitting criterion $\left|\frac{\tilde{\eta}}{d}\right|>k_{nh} = 0.001$. The blue area depicts the hydrostatic area, while the red represents the non-hydrostatic domain, where we apply the correction.