High Order Still-Water and Moving-Water Equilibria Preserving Discontinuous Galerkin Methods for the Ripa Model

Abstract

Shallow water equations with horizontal temperature gradients, also known as the Ripa system, are used to model flows when the temperature fluctuations play an important role. These equations admit steady state solutions where the fluxes and source terms balance each other. We present well-balanced discontinuous Galerkin methods for the Ripa model which can preserve the still-water or the general moving-water equilibria. The key ideas are the recovery of well-balanced states, separation of the solution into the equilibrium and fluctuation components, and appropriate approximations of the numerical fluxes and source terms. The same framework is also extended to design well-balanced methods for the constant height and isobaric steady state solutions of the Ripa model. Numerical examples are presented to verify the well-balanced property, high order accuracy, and good resolution for both smooth and discontinuous solutions.