Analysis of a regularized, time-staggered discretization method and its link to the semi-implicit method

A key aspect of the recently proposed Hamiltonian Particle-Mesh (HPM) method is its time-staggered discretization combined with a regularization of the continuous governing equations. In this paper, the time discretization aspect of the HPM method is analysed for the linearized, rotating, shallow-water equations with orography and the combined effect of time-staggering and regularization is compared analytically to the popular semi-implicit time discretization of the unregularized equations. It is found that the two approaches are essentially equivalent provided the regularization parameter is chosen appropriately in terms of the time step ∆t. The paper treats space as a continuum and, hence, its analysis is not limited to the HPM method.