An automatically well-balanced formulation of pressure forcing for discontinuous Galerkin methods for the shallow water equations

Abstract

This paper begins a development of methods for addressing variable bottom topography in discontinuous Galerkin numerical methods for multi-layer, variable-density models of ocean circulation. For numerical models of ocean circulation, it is a widespread practice to split the fast (external) and slow (internal) dynamics into separate subsystems that are solved by different techniques. The fast dynamics are modeled by a vertically-integrated system that is very similar to the shallow water equations for a hydrostatic fluid of constant density. As a first step, the present paper focuses on variable bottom topography for the shallow water equations; extensions to the multi-layer case will be reported elsewhere.A central point of this work, for both the shallow water and the multi-layer cases, is the representation of the pressure forcing in the momentum equations. For the shallow water case, the present work does not use the standard representation of the pressure forcing that is widely used for the shallow water system. Instead, it begins with a more basic form of the momentum equations for a fluid flow, and it proceeds directly to a weak Galerkin form via integration over a suitable fluid region. The resulting formulation of the momentum equations is automatically well-balanced, subject to an assumption that the algorithms used to compute quantities at cell edges reproduce the continuous values in the case where all functions involved are continuous. In addition, this formulation has a structure that is analogous to that of the momentum equations in the individual layers of a multi-layer fluid, and this facilitates consistency between the two subsystems that are used to model such a fluid.The numerical computations described here include tests of well-balancing and tests of wave propagation over variable topography.