Well-Balanced Finite-Volume Model for Long-Wave Runup

Abstract

This paper presents a two-dimensional, well-balanced finite-volume model for runup of long waves under nonbreaking and breaking conditions. The model uses a conservative form of the nonlinear shallow-water equations with source terms and an explicit Godunov-type scheme along with the exact Riemann solver for the flux and moving waterline. A second-order scheme splits the two-dimensional problem into two sequential one-dimensional problems for time integration. The surface-gradient method leads to a well-balanced formulation of the flux and source terms and a piecewise linear interpolation reconstructs numerical data at cell interfaces to achieve second-order accuracy in space. This provides accurate descriptions of the conserved variables and small flow-depth perturbations near the moving waterline. The computed surface elevation, flow velocity, and runup show very good agreement with previous asymptotic and analytical solutions as well as laboratory data. The model accurately describes breaking waves as bores or hydraulic jumps and conserves volume across flow discontinuities.