A high‐order triangular discontinuous Galerkin oceanic shallow water model

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. Copyright © 2007 John Wiley & Sons, Ltd.     

Semi‐Lagrangian semi‐implicit time‐splitting scheme for the shallow water equations

Time‐splitting technique applied in the context of the semi‐Lagrangian semi‐implicit method allows the use of extended time steps mainly based on physical considerations and reduces the number of numerical operations at each time step such that it is approximately proportional to the number of the points of spatial grid. To control time growth of the additional truncation errors, the standard stabilizing correction method is modified with no penalty for accuracy and efficiency of the algorithm. A linear analysis shows that constructed scheme is stable for time steps up to 2h. Numerical integrations with actual atmospheric fields of pressure and wind confirm computational efficiency, extended stability and accuracy of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

An unconditionally stable semi‐Lagrangian method for the spherical atmospherical shallow water equations

A semi‐implicit, semi‐Lagrangian, mixed finite difference–finite volume model for the shallow water equations on a rotating sphere is introduced and discussed. Its main features are the vectorial treatment of the momentum equation and the finite volume approach for the continuity equation. Pressure and Coriolis terms in the momentum equation and velocity in the continuity equation are treated semi‐implicitly. Moreover, a splitting technique is introduced to preserve symmetry of the numerical scheme. An alternative asymmetric scheme (without splitting) is also introduced and the efficiency of both is discussed. The model is shown to be conservative in geopotential height and unconditionally stable for 0.5≤θ≤1. Numerical experiments on two standard test problems confirm the performance of the model. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order accurate p‐multigrid discontinuous Galerkin solution of the Euler equations

Discontinuous Galerkin (DG) methods have proven to be perfectly suited for the construction of very high‐order accurate numerical schemes on arbitrary unstructured and possibly nonconforming grids for a wide variety of applications, but are rather demanding in terms of computational resources. In order to improve the computational efficiency of this class of methods a p‐multigrid solution strategy has been developed, which is based on a semi‐implicit Runge–Kutta smoother for high‐order polynomial approximations and the implicit Backward Euler smoother for piecewise constant approximations. The effectiveness of the proposed approach is demonstrated by comparison with p‐multigrid schemes employing purely explicit smoothing operators for several 2D inviscid test cases. Copyright © 2008 John Wiley & Sons, Ltd.     

An hp‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p?1, we allow both for the mesh size, h, and polynomial approximation order to dynamically change during the simulation. For the h‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p‐adaptivity is more efficient than h‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.     

Boundary discretization for high‐order discontinuous Galerkin computations of tidal flows around shallow water islands

In this paper some preliminary results concerning the application of the high‐order discontinuous Galerkin (DG) method for the resolution of realistic problems of tidal flows around shallow water islands are presented. In particular, tidal flows are computed around the Rattray island located in the Great Barrier Reef. This island is a standard benchmark problem well documented in the literature providing useful in situ measurements for validation of the model. Realistic elements of the simulation are a tidal flow forcing, a variable bathymetry and a non‐trivial coastline. The computation of tidal flows in shallow water around an island is very similar to the simulation of the Euler equations around bluff bodies in quasi‐steady flows. The main difference lies in the high irregularity of islands' shapes and in the fact that, in the framework of large‐scale ocean models, the number of elements to represent an island is drastically limited compared with classical engineering computations. We observe that the high‐order DG method applied to shallow water flows around bluff bodies with poor linear boundary representations produces oscillations and spurious eddies. Surprisingly those eddies may have the right size and intensity but may be generated by numerical diffusion and are not always mathematically relevant. Although not interested in solving accurately the boundary layers of an island, we show that a high‐order boundary representation is mandatory to avoid non‐physical eddies and spurious oscillations. It is then possible to parametrize accurately the subgrid‐scale processes to introduce the correct amount of diffusion in the model. The DG results around the Rattray island are eventually compared with current measurements and reveal good agreement. Copyright © 2008 John Wiley & Sons, Ltd.     

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

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary order for the spatial discretization and a third‐order Runge–Kutta scheme to advance in time. The local elements are coupled together by numerical fluxes, evaluated using the HLLC Riemann solver. We apply the model to test cases involving smooth flows and demonstrate the exponentially fast convergence with regard to polynomial order. We also illustrate that even for results of ‘engineering accuracy’ the computational efficiency increases with increasing order of the model and time of integration. The model is found to be robust in the presence of shocks where Gibbs oscillations can be suppressed by slope limiting. Copyright 2004 John Wiley & Sons, Ltd.     

Stability/dispersion analysis of the discontinuous Galerkin linearized shallow‐water system

The frequency or dispersion relation for the discontinuous Galerkin mixed formulation of the 1‐D linearized shallow‐water equations is analysed, using several basic DG mixed schemes. The dispersion properties are compared analytically and graphically with those of the mixed continuous Galerkin formulation for piecewise‐linear bases on co‐located grids. Unlike the Galerkin case, the DG scheme does not exhibit spurious stationary pressure modes. However, spurious propagating modes have been identified in all the present discontinuous Galerkin formulations. Numerical solutions of a test problem to simulate fast gravity modes illustrate the theoretical results and confirm the presence of spurious propagating modes in the DG schemes. Copyright © 2005 John Wiley & Sons, Ltd.     

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd.     

High‐order continuous and discontinuous Galerkin methods for wave problems

Three Galerkin methods—continuous Galerkin, Compact Discontinuous Galerkin, and hybridizable discontinuous Galerkin—are compared in terms of performance and computational efficiency in 2‐D scattering problems for low and high‐order polynomial approximations. The total number of DOFs and the total runtime are used for this correlation as well as the corresponding precision. The comparison is carried out through various numerical examples. The superior performance of high‐order elements is shown. At the same time, similar capabilities are shown for continuous Galerkin and hybridizable discontinuous Galerkin, when high‐order elements are adopted, both of them clearly outperforming compact discontinuous Galerkin. Copyright © 2013 John Wiley & Sons, Ltd.     

Practical evaluation of five partly discontinuous finite element pairs for the non‐conservative shallow water equations

This paper provides a comparison of five finite element pairs for the shallow water equations. We consider continuous, discontinuous and partially discontinuous finite element formulations that are supposed to provide second‐order spatial accuracy. All of them rely on the same weak formulation, using Riemann solver to evaluate interface integrals. We define several asymptotic limit cases of the shallow water equations within their space of parameters. The idea is to develop a comparison of these numerical schemes in several relevant regimes of the subcritical shallow water flow. Finally, a new pair, using non‐conforming linear elements for both velocities and elevation (P?P), is presented, giving optimal rates of convergence in all test cases. P?P₁ and P?P₁ mixed formulations lack convergence for inviscid flows. P?P₂ pair is more expensive but provides accurate results for all benchmarks. P?P provides an efficient option, except for inviscid Coriolis‐dominated flows, where a small lack of convergence is observed. Copyright © 2009 John Wiley & Sons, Ltd.     

An adaptive discretization of shallow‐water equations based on discontinuous Galerkin methods

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. Copyright © 2006 John Wiley & Sons, Ltd.     

Baroclinic stability for a family of two‐level,semi‐implicit numerical methods for the 3D shallow water equations

The baroclinic stability of a family of two time‐level, semi‐implicit schemes for the 3D hydrostatic, Boussinesq Navier–Stokes equations (i.e. the shallow water equations), which originate from the TRIM model of Casulli and Cheng (Int. J. Numer. Methods Fluids 1992; 15 :629–648), is examined in a simple 2D horizontal–vertical domain. It is demonstrated that existing mass‐conservative low‐dissipation semi‐implicit methods, which are unconditionally stable in the inviscid limit for barotropic flows, are unstable in the same limit for baroclinic flows. Such methods can be made baroclinically stable when the integrated continuity equation is discretized with a barotropically dissipative backwards Euler scheme. A general family of two‐step predictor‐corrector schemes is proposed that have better theoretical characteristics than existing single‐step schemes. Copyright © 2006 John Wiley & Sons, Ltd.     

An unstructured grid,three‐dimensional model based on the shallow water equations

A semi‐implicit finite difference model based on the three‐dimensional shallow water equations is modified to use unstructured grids. There are obvious advantages in using unstructured grids in problems with a complicated geometry. In this development, the concept of unstructured orthogonal grids is introduced and applied to this model. The governing differential equations are discretized by means of a semi‐implicit algorithm that is robust, stable and very efficient. The resulting model is relatively simple, conserves mass, can fit complicated boundaries and yet is sufficiently flexible to permit local mesh refinements in areas of interest. Moreover, the simulation of the flooding and drying is included in a natural and straightforward manner. These features are illustrated by a test case for studies of convergence rates and by examples of flooding on a river plain and flow in a shallow estuary. Copyright © 2000 John Wiley & Sons, Ltd.     

An eigenvector‐based linear reconstruction approach for time stepping in discontinuous Galerkin scheme used to solve shallow water equations

Discontinuous Galerkin (DG) methods have shown promising results for solving the two‐dimensional shallow water equations. In this paper, the classical Runge–Kutta (RK) time discretisation is replaced by the eigenvector‐based reconstruction (EVR) that allows the second‐order time accuracy to be achieved within a single time‐stepping procedure. Moreover, the EVRDG approach yields stable solutions near drying and wetting fronts, whereas the classical RKDG approach yields instabilities. The proposed EVRDG technique is compared with the original RKDG approach on various test cases with analytical solutions. The EVRDG solutions are shown to be as accurate as those obtained with the RKDG scheme. Besides, the EVRDG scheme is 1.6 times faster than the RKDG method. Simulating dambreaks involving dry beds confirms that EVRDG scheme gives correct solutions, whereas the RKDG method yields instabilities. Copyright © 2012 John Wiley & Sons, Ltd.     

An implicit time‐marching algorithm for shallow water models based on the generalized wave continuity equation

Wave equation models currently discretize the generalized wave continuity equation with a three‐time‐level scheme centered at k and the momentum equation with a two‐time‐level scheme centered at k+1/2; non‐linear terms are evaluated explicitly. However in highly non‐linear applications, the algorithm becomes unstable at even moderate Courant numbers. This paper examines an implicit treatment of the non‐linear terms using an iterative time‐marching algorithm. Depending on the domain, results from one‐dimensional experiments show up to a tenfold increase in stability and temporal accuracy. The sensitivity of stability to variations in the G‐parameter (a numerical weighting parameter in the generalized wave continuity equation) was examined; results show that the greatest increase in stability occurs with G/τ=2–50. In the one‐dimensional experiments, three different types of node spacing techniques—constant, variable, and LTEA (Localized Truncation Error Analysis)—were examined; stability is positively correlated to the uniformity of the node spacing. Lastly, a scaling analysis demonstrates that the magnitudes of the non‐linear terms are positively correlated to those that most influence stability, particularly the term containing the G‐parameter. It is evident that the new algorithm improves stability and temporal accuracy in a cost‐effective manner. Copyright © 2001 John Wiley & Sons, Ltd.     

An improvement of classical slope limiters for high‐order discontinuous Galerkin method

In this paper, we describe some existing slope limiters (Cockburn and Shu's slope limiter and Hoteit's slope limiter) for the two‐dimensional Runge–Kutta discontinuous Galerkin (RKDG) method on arbitrary unstructured triangular grids. We describe the strategies for detecting discontinuities and for limiting spurious oscillations near such discontinuities, when solving hyperbolic systems of conservation laws by high‐order discontinuous Galerkin methods. The disadvantage of these slope limiters is that they depend on a positive constant, which is, for specific hydraulic problems, difficult to estimate in order to eliminate oscillations near discontinuities without decreasing the high‐order accuracy of the scheme in the smooth regions. We introduce the idea of a simple modification of Cockburn and Shu's slope limiter to avoid the use of this constant number. This modification consists in: slopes are limited so that the solution at the integration points is in the range spanned by the neighboring solution averages. Numerical results are presented for a nonlinear system: the shallow water equations. Four hydraulic problems of discontinuous solutions of two‐dimensional shallow water are presented. The idealized dam break problem, the oblique hydraulic jump problem, flow in a channel with concave bed and the dam break problem in a converging–diverging channel are solved by using the different slope limiters. Numerical comparisons on unstructured meshes show a superior accuracy with the modified slope limiter. Moreover, it does not require the choice of any constant number for the limiter condition. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

This paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.     

A third‐order implicit discontinuous Galerkin method based on a Hermite WENO reconstruction for time‐accurate solution of the compressible Navier–Stokes equations

A space and time third‐order discontinuous Galerkin method based on a Hermite weighted essentially non‐oscillatory reconstruction is presented for the unsteady compressible Euler and Navier–Stokes equations. At each time step, a lower‐upper symmetric Gauss–Seidel preconditioned generalized minimal residual solver is used to solve the systems of linear equations arising from an explicit first stage, single diagonal coefficient, diagonally implicit Runge–Kutta time integration scheme. The performance of the developed method is assessed through a variety of unsteady flow problems. Numerical results indicate that this method is able to deliver the designed third‐order accuracy of convergence in both space and time, while requiring remarkably less storage than the standard third‐order discontinous Galerkin methods, and less computing time than the lower‐order discontinous Galerkin methods to achieve the same level of temporal accuracy for computing unsteady flow problems. Copyright © 2015 John Wiley & Sons, Ltd.