Boundary discretization for high-order discontinuous Galerkin computations of tidal flows around shallow water islands (original) (raw)
A high-order triangular discontinuous Galerkin oceanic shallow water model
International Journal for Numerical Methods in Fluids, 2008
A high-order triangular discontinuous Galerkin (DG) method is applied to the two-dimensional oceanic shallow water equations. The DG method can be characterized as the fusion of finite elements with finite volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however, by using straight-edged triangles we eliminate the mass matrix completely from the discrete equations. Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on testing the discrete spatial operators by using six test cases-three of which have analytic solutions. The three tests having analytic solutions show that the high-order triangular DG method exhibits exponential convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior for all polynomial orders and test cases considered.
A discontinuous Galerkin method for two-dimensional flow and transport in shallow water
Advances in Water Resources, 2002
A discontinuous Galerkin (DG) finite element method is described for the two-dimensional, depth-integrated shallow water equations (SWEs). This method is based on formulating the SWEs as a system of conservation laws, or advection-diffusion equations. A weak formulation is obtained by integrating the equations over a single element, and approximating the unknowns by piecewise, possibly discontinuous, polynomials. Because of its local nature, the DG method easily allows for varying the polynomial order of approximation. It is also ''locally conservative'', and incorporates upwinded numerical fluxes for modeling problems with high flow gradients. Numerical results are presented for several test cases, including supercritical flow, river inflow and standard tidal flow in complex domains, and a contaminant transport scenario where we have coupled the shallow water flow equations with a transport equation for a chemical species. Ó : S 0 3 0 9 -1 7 0 8 ( 0 1 ) 0 0 0 1 9 -7
Space-time discontinuous Galerkin discretization of rotating shallow water equations on moving grids
2006
A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova's discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge-Kutta method. A thorough verification of the space-time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore-vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space-time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid.
hp Discontinuous Galerkin methods for advection dominated problems in shallow water flow
Computer Methods in Applied Mechanics and Engineering, 2006
In this paper, we discuss the development, verification, and application of an hp discontinuous Galerkin (DG) finite element model for solving the shallow water equations (SWE) on unstructured triangular grids. The h and p convergence properties of the method are demonstrated for both linear and highly nonlinear problems with advection dominance. Standard h-refinement for a fixed p leads to p + 1 convergence rates, while exponential convergence is observed for p-refinement for a fixed h. It is also demonstrated that the use of prefinement is more efficient for problems exhibiting smooth solutions. Additionally, the ability of p-refinement to adequately resolve complex, two-dimensional flow structures is demonstrated in the context of a coastal inlet problem.
Space–time discontinuous Galerkin discretization of rotating shallow water equations
Journal of Computational Physics, 2007
A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova's discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge-Kutta method. A thorough verification of the space-time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one-dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore-vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space-time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid.
An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods
International journal for …, 2006
In this paper, we present a Discontinuous Galerkin formulation of the shallow water equations. An orthogonal basis is used for the spatial discretization and an explicit Runge-Kutta scheme is used for time discretization. Some results of second order anisotropic adaptive calculations are presented for dam breaking problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities like hydraulic jumps.
A three-dimensional finite-element model is used to investigate the tidal flow around Rattray Island, Great Barrier Reef, Australia. Field measurements and visual observations show both stable eddies developing at rising and falling tide in the wake of the island. The water turbidity suggests intense upwelling able to carry bed sediments upwards. Based on previous numerical studies, it remains unclear at this point whether the most intense upwelling occurs near the centre of the eddies or off the island's tips, closer to the island. All these studies resorted to a very simple turbulence closure, with a zeroequation model whereby the coefficient of vertical viscosity is computed via an algebraic expression. In this work, we aim at studying the influence of the turbulence closure on model results, with emphasis on the prediction of vertical motions. The Mellor and Yamada level 2.5 closure scheme is used and an increase in the intensity of vertical transport is observed. This increase is partly explained by the fact that the Mellor and Yamada model takes into account the hysteresis effect in the time variation of turbulence variables. The influence of the advection of turbulence variables is estimated to be negligible. By a better representation of transient coastal phenomena, the Mellor and Yamada level 2.5 turbulence closure improves the model to a significant degree. r
A BGK-Based Discontinuous Galerkin Method for the Navier-Stokes Equations on Arbitrary Grids
46th AIAA Aerospace Sciences Meeting and Exhibit, 2008
A discontinuous Galerkin Method based on a Bhatnagar-Gross-Krook (BGK) formulation is presented for the solution of the compressible Navier-Stokes equations on arbitrary grids. The idea behind this approach is to combine the robustness of the BGK scheme with the accuracy of the DG methods in an effort to develop a more accurate, efficient, and robust method for numerical simulations of viscous flows in a wide range of flow regimes. Unlike the traditional discontinuous Galerkin methods, where a Local Discontinuous Galerkin (LDG) formulation is usually used to discretize the viscous fluxes in the Navier-Stokes equations, this DG method uses a BGK scheme to compute the fluxes which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. The developed method is used to compute a variety of viscous flow problems on arbitrary grids. The numerical results obtained by this BGKDG method are extremely promising and encouraging in terms of both accuracy and robustness, indicating its ability and potential to become not just a competitive but simply a superior approach than the current available numerical methods.
A discontinuous Galerkin method for two-layer shallow water equations
Mathematics and Computers in Simulation, 2015
In this paper, we study a discontinuous Galerkin method to approximate solutions of the two-layer shallow water equations on non-flat topography. The layers can be formed in the shallow water model based on the vertical variation of water density which in general depends on the water temperature and salinity. For a water body with equal density the model reduces to the canonical single-layer shallow water equations. Thus, for a model with equal density on flat bottom, the method is equivalent to the discontinuous Galerkin method for conservation laws. The considered method is a stable, highly accurate and locally conservative finite element method whose approximate solutions are discontinuous across inter-element boundaries; this property renders the method ideally suited for the hpadaptivity. Several numerical results illustrate the performance of the method and confirm its capability to solve two-layer shallow water flows including tidal conditions on the water free-surface and bed frictions on the bottom topography.
International Journal for Numerical Methods in Fluids, 2012
This paper presents multirate explicit time stepping schemes for solving Partial Differential Equations with Discontinuous Galerkin (DG) elements in the framework of large-scale marine flows. It addresses the variability of the local stable time steps by gathering the mesh elements in appropriate groups. The real challenge consists to develop methods exhibiting mass conservation and consistency. Two multirate approaches, based on standard Explicit Runge-Kutta methods, are analyzed. They are well suited and optimized for the DG framework. The significant speedups observed for the hydrodynamic application of the Great Barrier Reef confirm the theoretical expectations.