Emerging Numerical Methods for Atmospheric Modeling

Abstract

This chapter discusses the development of discontinuous Galerkin (DG) schemes for the hyperbolic conservation laws relevant to atmospheric modeling. Two variants of the DG spatial discretization, the modal and nodal form, are considered for the one- and two-dimensional cases. The time integration relies on a second- or third-order explicit strong stability-preserving Runge–Kutta method. Several computational examples are provided, including a solid-body rotation test, a deformational flow problem and solving the barotropic vorticity equation for an idealized cyclone. A detailed description of various limiters available for the DG method is given, and a new limiter with positivity-preservation as a constraint is proposed for two-dimensional transport. The DG method is extended to the cubed-sphere geometry and the transport and shallow water models are discussed.