High-order semi-implicit time-integration of a triangular discontinuous Galerkin oceanic shallow water model

We extend the explicit in time high-order triangular discontinuous Galerkin (DG) method to semi-implicit and then apply the algorithm to the two-dimensional oceanic shallow water equations; we implement high-order semi-implicit time-integrators using the backward difference formulas from orders one through six. The reason for changing the time-integration method from explicit to semi-implicit is that the explicit method requires a very small time-step in order to maintain stability, especially for high-order DG methods. Changing the time-integration method to be semi-implicit allows one to circumvent the stability criterion due to the gravity waves, which for most shallow water applications are the fastest waves in the system (the exception being supercritical flow where the Froude number is greater than one). The challenge of constructing a semi-implicit method for a DG model is that the DG machinery requires not only the standard finite element-type (FE) area integrals but finite volume-type (FV) boundary integrals as well. These boundary integrals pose the biggest challenge in a semi-implicit discretization because they require the construction of a Riemann solver that is the true linear representation of the nonlinear Riemann problem; if this condition is not satisfied then the resulting numerical method will not be consistent with the continuous equations. In this paper we present semi-implicit time-integrators for the DGmethod that maintain most of the usual attribute