A new Poisson-type equation applicable to the three-dimensional non-hydrostatic model in the framework of the discontinuous Galerkin method

Abstract

In this study, a new Poisson-type equation for non-hydrostatic pressure in terrain-following σ-coordinates is proposed. The resulting equation consists of a standard elliptic operator and other first-order operators for non-hydrostatic pressure. Compared with the original Poisson-type equation, the treatment of the boundary condition at the lateral boundary can be simplified. Numerical tests indicate that the reformulated Poisson-type equation shows better convergence properties than the original equation when the quadrature-free nodal discontinuous Galerkin method is adopted for numerical discretization. Additionally, for the standing wave case, the numerical error of the three-dimensional non-hydrostatic model with the reformulated Poisson equation is smaller than that with the original equation, the numerical results of the solitary wave case indicate that the dissipation of the three-dimensional non-hydrostatic model is small, and transformation of wave over a circular shoal indicates that the developed model can properly simulate combined wave refraction and diffraction.