Construction and Analysis of HDG Methods for Linearized Shallow Water Equations

Abstract

We present a systematic and constructive methodology to devise various hybridized discontinuous Galerkin (HDG) methods for linearized shallow water equations. It is shown that using the Rankine--Hugoniot condition to solve the Riemann problem is a natural approach to deriving HDG methods. At the heart of our development is an upwind HDG framework obtained by hybridizing the upwind flux in the standard discontinuous Galerkin (DG) approach. Essentially, the HDG framework is a redesign of the standard DG approach to reducing the number of coupled unknowns. An upwind and three other HDG methods are constructed and analyzed for linearized shallow water systems. Rigorous stability and convergence analysis for both semidiscrete and fully discrete systems are provided. We extend the upwind HDG method to a family of penalty HDG schemes and rigorously analyze their well-posedness, stability, and convergence rates. Numerical results for the linear standing wave and the Kelvin wave for oceanic shallow water systems are presented to verify our theoretical findings.