Numerical dispersion in non-hydrostatic modeling of long-wave propagation

Abstract

Numerical discretization with a finite-difference scheme is known to introduce frequency dispersion in depth-integrated long-wave models commonly used in tsunami research and hazard mapping. While prior studies on numerical dispersion focused on the linear shallow-water equations, we include the non-hydrostatic pressure and vertical velocity through a Keller box scheme and investigate the properties of the resulting system in relation to a hydrostatic model. Fourier analysis of the discretized governing equations gives rise to a dispersion relation in terms of the water-depth parameter as well as the time step, grid size, and wave direction. The dispersion relation is illustrated by its lead-order approximation derived from Taylor series expansions as well as numerical experiments involving standing and progressive waves with uniform water depth. Interaction between the spatial discretization and non-hydrostatic terms leads to significant reduction of numerical dispersion outside the shallow-water range. Numerical dispersion also decreases for wave propagation oblique to the computational grid due to effective increase in spatial resolution. The time step, which counteracts numerical dispersion from spatial discretization, only has secondary effects within the applicable range of Courant numbers. Since the governing equations of the non-hydrostatic model derived from the Keller box scheme tend to underestimate dispersion in shoaling water, the numerical effects are complementary in producing a solution closer to Airy wave theory. A case study of the 2011 Tohoku tsunami illustrates the grid sensitivity and convergence properties in real-world applications. A properly selected grid size can achieve an accurate description of wave propagation over a wide range of water-depth parameters across the ocean.