  High order schemes are known to be unstable in the presence of shock
discontinuities or under-resolved solution features, and have traditionally
required additional filtering, limiting, or artificial viscosity to avoid
solution blow up. Entropy stable schemes address this instability by ensuring
that physically relevant solutions satisfy a semi-discrete entropy inequality
independently of discretization parameters. However, additional measures must
be taken to ensure that solutions satisfy physical constraints such as
positivity. In this work, we present a high order entropy stable discontinuous
Galerkin (ESDG) method for the nonlinear shallow water equations (SWE) on
two-dimensional (2D) triangular meshes which preserves the positivity of the
water heights. The scheme combines a low order positivity preserving method
with a high order entropy stable method using convex limiting. This method is
entropy stable and well-balanced for fitted meshes with continuous bathymetry
profiles.
