Conservative upwind difference schemes for the shallow water equations

A study of a number of current numerical schemes for the shallow water equations leads to the establishment of relationships between these schemes. Further analysis then suggests new formulations of the schemes, as well as an alternative scheme having the same key properties.     

An adaptive solver for the spherical shallow water equations

The shallow water equations (SWE), which describe the flow of a thin layer of fluid in two dimensions have been used by the atmospheric modelling community as a vehicle for testing promising numerical methods for solving atmospheric and oceanic problems. The SWE are important for the study of the dynamics of large-scale flows, as well for the development of new numerical schemes that are applied to more complex models. In this paper we present a finite difference p-adaptive method based on high order finite differences that is applied using an error indicator for solving the SWE on the sphere. A standard test set is used to evaluate the accuracy of the new method. The results obtained are compared with the pseudo-spectral method.     

A discontinuous Galerkin algorithm for the two-dimensional shallow water equations

A new high-resolution finite element scheme is introduced for solving the two-dimensional (2D) depth-integrated shallow water equations (SWE) via local plane approximations to the unknowns. Bed topography data are locally approximated in the same way as the flow variables to render an instinctive well-balanced scheme. A finite volume (FV) wetting and drying technique that reconstructs the Riemann states by ensuring non-negative water depth and maintaining well-balanced solution is adjusted and implemented in the current finite element framework. Meanwhile, a local slope-limiting process is applied and those troubled-slope-components are restricted by the minmod FV slope limiter. The inter-cell fluxes are upwinded using the HLLC approximate Riemann solver. Friction forces are separately evaluated via stable implicit discretization to the finite element approximating coefficients. Boundary conditions are derived and reported in details. The present model is validated against several test cases including dam-break flows on regular and irregular domains with flooding and drying.     

Lattice Boltzmann method for variable density shallow water equations

A lattice Boltzmann method is developed for solution of a form of the shallow water equations that is suitable for flows which are fully mixed in the vertical direction but have variable density in the horizontal plane. In the present approach, double distribution functions are applied: one for the shallow water flows and the other for the mass transport. Direct coupling between the water flow and mass transport is achieved by updating the flow density from the concentration during simulation. Accuracy and applicability of the model are demonstrated by two numerical tests: the stationary hydrostatic equilibrium of liquid of variable density in a tank with non-uniform bed terrain, and the horizontal diffusion of species with an initial Gaussian distribution of concentration in a uniform flow field.     

A consistent control of spurious singular modes in the 9-node Lagrange element for the laplace and mindlin plate equations

When 2 × 2 quadrature is used with the 9-node Lagrange element, which is essential in C⁰ plate elements to avoid locking, spurious singular modes appear on the element level which can lead to singularity or near-singularity of the global equations. Here these modes are controlled by a procedure that introduces additional generalized stresses and strains so that the spurious modes are eliminated and the consistency of the resulting finite difference equations is not impaired; hence the procedure passes the patch test. Applications to the diffusion and Mindlin plate equations are presented. Results show that h³ convergence in the L₂-norm is almost retained.     

Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations

The Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements.Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations.Two types of Lagrange-Galerkin methods are presented: the strong and weak formulations. The strong form relies mainly on interpolation to achieve high accuracy while the weak form relies primarily on integration. Lagrange-Galerkin schemes offer an increased efficiency by virtue of their less stringent CFL condition. The use of quadrilateral elements permits the construction of spectral-type finite-element methods that exhibit exponential convergence as in the conventional spectral method, yet they are constructed locally as in the finite-element method; this is the spectral method.In this paper, we show how to fuse the Lagrange-Galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes.     

Construction of solutions for the shallow water equations by the decomposition method

This paper deals with the implementation of Adomian’s decomposition method for the variable-depth shallow water equations with source term. Using this method, the solutions were calculated in the form of a convergent power series with easily computable components. The convergence of the method is illustrated numerically.     

The penalty method for the Navier-Stokes equations

Summary The penalty finite element method as it applies to the Stokes and Navier-Stokes flow equations is reviewed. The main developments are discussed and selected but still extensive list of references is provided.     

A comparison of conservative upwind difference schemes for the shallow water equations

Numerical results are presented and compared for four conservative upwind difference schemes for the shallow water equations when applied to a standard test problem. This includes consideration of the effect of treating part of the flux balance as a source, and a comparison of square-root and arithmetic averaging.     

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.     

Quadratic spline methods for the shallow water equations on the sphere: Collocation

In this study, we present numerical methods, based on the optimal quadratic spline collocation (OQSC) methods, for solving the shallow water equations (SWEs) in spherical coordinates. The error associated with quadratic spline interpolation is fourth order locally at certain points and third order globally, but the standard quadratic spline collocation methods generate only second-order approximations. In contrast, the OQSC methods generate approximations of the same order as quadratic spline interpolation. In the one-step OQSC method, the discrete differential operators are perturbed to eliminate low-order error terms, and a high-order approximation is computed using the perturbed operators. In the two-step OQSC method, a second-order approximation is generated first, using the standard formulation, and then a high-order approximation is computed in a second phase by perturbing the right sides of the equations appropriately. In this implementation, the SWEs are discretized in time using the semi-Lagrangian semi-implicit method, and in space using the OQSC methods. The resulting methods are efficient and yield stable and accurate representation of the meteorologically important Rossby waves. Moreover, by adopting the Arakawa C-type grid, the methods also faithfully capture the group velocity of inertia-gravity waves.     

A nonlinear Galerkin method for the Navier-Stokes equations

Modern large scale computing allows the utilization of a very large number of variables/modes for spatial discretization. Therefore the computer tends to be saturated by computations on small wavelengths that carry a small percentage of the total energy. We advocate the utilization of algorithms treating differently small wavelengths and large wavelengths and we present here an algorithm of this sort, the nonlinear Galerkin method, stemming from the dynamical system theory.     

The bid method for the steady-state Navier-Stokes equations

The Biharmonic Driver (BID) method uses direct (non-iterative) linear biharmonic solvers to obtain solutions to the nonlinear two-dimensional steady-state incompressible Navier-Stokes equations. Low Reynolds number recirculating flows are solved in 6–8 non-time-like iterations. Modifications are suggested for flow-through problems, and comparisons are made with other semidirect methods.     

Penalty finite element method for the Navier-Stokes equations

We present an analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(ε) where ε → 0 is the penalty parameter.Existence, uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized Navier-Stokes problem presented here. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.     

A mimetic mass, momentum and energy conserving discretization for the shallow water equations

A spatial semi-discretization is developed for the two-dimensional depth-averaged shallow water equations on a non-equidistant structured and staggered grid. The vector identities required for energy conservation in the continuous case are identified. Discrete analogues are developed, which lead to a finite-volume semi-discretisation which conserves mass, momentum, and energy simultaneously. The key to discrete energy conservation for the shallow water equations is to numerically distinguish storage of momentum from advective transport of momentum. Simulation of a large-amplitude wave in a basin confirms the conservative properties of the new scheme, and demonstrates the enhanced robustness resulting from the compatibility of continuity and momentum equations. The scheme can be used as a building block for constructing fully conservative curvilinear, higher order, variable density, and non-hydrostatic discretizations.     

The discontinuous Galerkin finite element method for the 2D shallow water equations

A high-order finite element method, total variational diminishing (TVD) Runge–Kutta discontinuous Galerkin method is investigated to solve free-surface problems in hydraulic dynamics. Some cases of circular dam and rapidly varying two-dimensional flows are presented to show the efficiency and stability of this method. The numerical simulations are given on structured rectangular mesh for regular domain and on unstructured triangular mesh for irregular domain, respectively.     

Finite element or finite difference methods for the two-dimensional shallow water equations?

The computational costs of some finite element and finite difference methods currently used to solve the shallow water equations are compared on theoretical grounds. It is shown that because band algorithms are employed, the finite element methods considered are not economically attractive for practical calculations. This weakness is not intrinsic to the finite element approach and ways in which it might be avoided are briefly discussed.     

An efficient unstructured MUSCL scheme for solving the 2D shallow water equations

The aim of this paper is to present a novel monotone upstream scheme for conservation law (MUSCL) on unstructured grids. The novel edge-based MUSCL scheme is devised to construct the required values at the midpoint of cell edges in a more straightforward and effective way compared to other conventional approaches, by making better use of the geometrical property of the triangular grids. The scheme is incorporated into a two-dimensional (2D) cell-centered Godunov-type finite volume model as proposed in Hou et al. (2013a,c) to solve the shallow water equations (SWEs). The MUSCL scheme renders the model to preserve the well-balanced property and achieve high accuracy and efficiency for shallow flow simulations over uneven terrains. Furthermore, the scheme is directly applicable to all triangular grids. Application to several numerical experiments verifies the efficiency and robustness of the current new MUSCL scheme.