Hybridized Discontinuous Galerkin Methods for Linearized Shallow Water Equations

We present a systematic and constructive methodology to devise various hybridized discontinuous Galerkin (HDG) methods for linearized shallow water equations. At the heart of our development is an upwind HDG framework obtained by hybridizing the upwind flux in the standard discontinuous Galerkin (DG) approach. The chief idea is to first break the uniqueness of the upwind flux across element boundaries by introducing a single-valued new trace unknowns on the mesh skeleton, and then re-enforce the uniqueness via algebraic conservation constraints. The beauty of this approach is that the actual globally coupled unknowns are those newly introduced trace unknowns, and hence the resulting matrix is substantially smaller and sparser. In particular, the usual DG unknowns can be solved for in an element-by-element manner, completely independent of each other, once the trace unknowns are computed. Essentially, the HDG framework is a redesign of the standard DG approach to reduce the number of coupled unknowns. The framework is first established for a general linear system of partial differential equations, and then applied to linearized shallow water systems. An upwind and three others HDG methods are constructed and analyzed. Rigorous stability and convergence analysis for both semi-discrete 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 as well. Numerical results for linear standing wave and Kelvin wave for oceanic shallow water systems are presented to verify our theoretical developments.