Discontinuous Galerkin discretization of shallow water equations in implicit primal formulations for turbulent stresses

Although the discontinuous Galerkin (DG) methods have been widely applied as an effective numerical tool for hyperbolic conservation equations (e.g., shallow water equations (SWEs) and compressible Navier-Stokes equations), one of the well-known drawbacks is the inconvenience in the treatment of second or higher derivative terms. For this reason, since the beginning of DG in the 1970s, many researchers have made efforts in devising accurate and consistent schemes to incorporate the possible jumps in solutions for higher-order derivatives. The turbulent stress terms of the SWEs are expressed as second-order derivatives and cannot be neglected due to their practical importance. So far, as a traditional approach, a flux formulation has been applied in DG SWE modeling. However, it is often criticized for being inefficient in terms of memory and computation time. In this study, the BR2 scheme, a well-known primal formulation in the DG computational fluid dynamics community, was employed and combined with the implicit Euler backward difference scheme for SWEs. The developed model was applied to four benchmark problems (including channel contraction and diverging, partial dam-break flow, and curved channel flow), and good agreements were observed.