Seminar: Numerical Dispersion in Non-Hydrostatic Modeling of Tsunami Propagation

Linyan Li, PhD
Post-Doctoral Research Fellow
Department of Ocean and Resources Engineering, University of Hawai’i at Mānoa

Numerical discretization with a finite-difference scheme is known to introduce frequency dispersion in depth-integrated 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 time step, grid size, wave direction, and water depth. The dispersion relation is illustrated by its lead-order approximation derived by the Taylor series expansion, numerical experiments involving standing and progressive waves with uniform water depth, and a sensitivity analysis of the grid size in modeling of the 2011 Tohoku tsunami across the Pacific Ocean. Interaction between the spatial discretization and non-hydrostatic terms results in 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 properly selected grid size can achieve accurate description of wave propagation over a wide range of water depth parameters.