A conservative staggered discontinuous Galerkin method on triangular grids for the shallow water equations

In this paper, we present a novel staggered discontinuous Galerkin (SDG) method on general staggered triangular grids for solving the shallow water equations arising in planetary geophysical flows. In the proposed spatial discretization with SDG, the normal component of the velocity function is designed to be continuous in each primal macro element and the fluid height continuous in each dual macro element. Our SDG method preserves key conservation properties, simultaneously ensuring local conservation of the mass and global conservation of the energy and vorticity. Dispersion analysis also reveals that our SDG method is suitable to suppress the potential spurious modes by maintaining an appropriate ratio between the degrees of freedom of the vorticity, velocity and fluid height variables. By further employing our SDG scheme alongside the mass-lumping finite element method for the vorticity equation, block diagonal mass matrices for velocity and height equations and diagonal mass matrices for vorticity equation are achieved, which can facilitate the inversion of mass matrices and consequently enhance the computational efficiency of explicit time stepping. Several benchmark experiments are also carried out to validate the proposed SDG method and demonstrate its conservation properties and efficiency.