A triangular spectral/ hp discontinuous Galerkin method for modelling 2D shallow water equations

114

110

SUMMARYWe build and analyze a Runge-Kutta Discontinuous Galerkin method to approximate the one-and two-dimensional Shallow-Water Equations. We introduce a flux modification technique to derive a wellbalanced scheme preserving steady-states at rest with variable bathymetry and a slope modification technique to deal satisfactorily with flooding and drying. Numerical results illustrating the performance of the proposed scheme are presented.

Section: Approximation Of Swe By Rkdg Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

Ern

Piperno

Djadel

2007

114

110

“…Dupont and Lin [5], Eskilsson and Sherwin [6], Remacle et al [7], and Kubatko et al [8] constructed shallow water models on triangles using a collapsed local coordinate DG method. Giraldo and Warburton [9] developed a HO DG oceanic shallow water model on unstructured adaptive triangular grids.…”

Section: Introductionmentioning

confidence: 99%

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

Giraldo

Restelli

2010

SUMMARYWe 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 timeintegration 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

Continuous, discontinuous and coupled discontinuous–continuous Galerkin finite element methods for the shallow water equations

Dawson

Westerink

Feyen

et al. 2006

SUMMARYWe consider the approximation of the depth-averaged two-dimensional shallow water equations by both a traditional continuous Galerkin (CG) ÿnite element method as well as two discontinuous Galerkin (DG) approaches. The DG method is locally conservative, ux-continuous on each element edge, and is suitable for both smooth and highly advective ows. A novel technique of coupling a DG method for continuity with a CG method for momentum is developed. This formulation is described in detail and validation via numerical testing is presented. Comparisons between a widely used CG approach, a conventional DG method, and the novel coupled discontinuous-continuous Galerkin method illustrates advantages and disadvantages in accuracy and e ciency.