High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model

• Published in: International journal for numerical methods in fluids Vol. 63; no. 9; pp. 1077 - 1102
• Main Authors: Giraldo, F. X., Restelli, M.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 30.07.2010; Wiley
• Subjects: Computational methods in fluid dynamics; discontinuous Galerkin; Discretization; Dynamics of the ocean (upper and deep oceans); Earth, ocean, space; Exact sciences and technology; explicit; External geophysics; finite element; finite volume; Fluid dynamics; Fundamental areas of phenomenology (including applications); Galerkin methods; implicit; Integrals; Mathematical analysis; Mathematical models; Physics; Physics of the oceans; semi-implicit; Shallow water; Spatial resolution; Stability; triangle; Discontinuous Galerkin method; implicit; Computational fluid dynamics; Viscous fluid; finite element; Finite volume method; digital simulation; Ocean dynamics; finite volume; finite element analysis; shallow water; Modeling; triangle; Rossby wave; explicit; Kelvin wave; discontinuous Galerkin; semi-implicit; Incompressible fluid; Shallow-water equations; Standing wave
• ISSN: 0271-2091; 1097-0363; 1097-0363
• DOI: 10.1002/fld.2118

We extend the explicit in time high‐order triangular discontinuous Galerkin (DG) method to semi‐implicit (SI) and then apply the algorithm to the two‐dimensional oceanic shallow water equations; we implement high‐order SI time‐integrators using the backward difference formulas from orders one to six. The reason for changing the time‐integration method from explicit to SI is that explicit methods require a very small time step in order to maintain stability, especially for high‐order DG methods. Changing the time‐integration method to SI 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 SI method for a DG model is that the DG machinery requires not only the standard finite element‐type area integrals, but also the finite volume‐type boundary integrals as well. These boundary integrals pose the biggest challenge in a SI 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 couple the SI time‐integrators with the DG method while maintaining most of the usual attributes associated with DG methods such as: high‐order accuracy (in both space and time), parallel efficiency, excellent stability, and conservation. The only property lost is that of a compact communication stencil typical of time‐explicit DG methods; implicit methods will always require a much larger communication stencil. We apply the new high‐order SI DG method to the shallow water equations and show results for many standard test cases of oceanic interest such as: standing, Kelvin and Rossby soliton waves, and the Stommel problem. The results show that the new high‐order SI DG model, that has already been shown to yield exponentially convergent solutions in space for smooth problems, results in a more efficient model than its explicit counterpart. Furthermore, for those problems where the spatial resolution is sufficiently high compared with the length scales of the flow, the capacity to use high‐order (HO) time‐integrators is a necessary complement to the employment of HO space discretizations, since the total numerical error would be otherwise dominated by the time discretization error. In fact, in the limit of increasing spatial resolution, it makes little sense to use HO spatial discretizations coupled with low‐order time discretizations. Published in 2009 by John Wiley & Sons, Ltd.