scholarly journals A Discontinuous Galerkin Global Shallow Water Model

2005 ◽  
Vol 133 (4) ◽  
pp. 876-888 ◽  
Author(s):  
Ramachandran D. Nair ◽  
Stephen J. Thomas ◽  
Richard D. Loft
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Time Integration ◽  
Curvilinear Coordinates ◽  
Basis Set ◽  
Water Model ◽  
Test Case ◽  
Shallow Water Model ◽  
Cubed Sphere

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. The continuous flux form nonlinear shallow water equations in curvilinear coordinates are employed. The spatial discretization employs a modal basis set consisting of Legendre polynomials. Fluxes along the element boundaries (internal interfaces) are approximated by a Lax–Friedrichs scheme. A third-order total variation diminishing Runge–Kutta scheme is applied for time integration, without any filter or limiter. Numerical results are reported for the standard shallow water test suite. The numerical solutions are very accurate, there are no spurious oscillations in test case 5, and the model conserves mass to machine precision. Although the scheme does not formally conserve global invariants such as total energy and potential enstrophy, conservation of these quantities is better preserved than in existing finite-volume models.

Diffusion Experiments with a Global Discontinuous Galerkin Shallow-Water Model

2009 ◽  
Vol 137 (10) ◽  
pp. 3339-3350 ◽  
Author(s):  
Ramachandran D. Nair
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Water Model ◽  
Order System ◽  
Small Scale ◽  
Shallow Water Model ◽  
First Order ◽  
Advection Diffusion Equation ◽  
Diffusion Scheme ◽  
Cubed Sphere

Abstract A second-order diffusion scheme is developed for the discontinuous Galerkin (DG) global shallow-water model. The shallow-water equations are discretized on the cubed sphere tiled with quadrilateral elements relying on a nonorthogonal curvilinear coordinate system. In the viscous shallow-water model the diffusion terms (viscous fluxes) are approximated with two different approaches: 1) the element-wise localized discretization without considering the interelement contributions and 2) the discretization based on the local discontinuous Galerkin (LDG) method. In the LDG formulation the advection–diffusion equation is solved as a first-order system. All of the curvature terms resulting from the cubed-sphere geometry are incorporated into the first-order system. The effectiveness of each diffusion scheme is studied using the standard shallow-water test cases. The approach of element-wise localized discretization of the diffusion term is easy to implement but found to be less effective, and with relatively high diffusion coefficients, it can adversely affect the solution. The shallow-water tests show that the LDG scheme converges monotonically and that the rate of convergence is dependent on the coefficient of diffusion. Also the LDG scheme successfully eliminates small-scale noise, and the simulated results are smooth and comparable to the reference solution.

A Two-Stage Fourth-Order Multimoment Global Shallow-Water Model on the Cubed Sphere

2020 ◽  
Vol 148 (10) ◽  
pp. 4267-4279
Author(s):  
Yuzhang Che ◽  
Chungang Chen ◽  
Feng Xiao ◽  
Xingliang Li ◽  
Xueshun Shen
Keyword(s):  
Shallow Water ◽  
Fourth Order ◽  
Time Integration ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step ◽  
Two Stage ◽  
Order Scheme ◽  
Runge Kutta ◽  
Cubed Sphere

AbstractA new multimoment global shallow-water model on the cubed sphere is proposed by adopting a two-stage fourth-order Runge–Kutta time integration. Through calculating the values of predicted variables at half time step t = tn + (1/2)Δt by a second-order formulation, a fourth-order scheme can be derived using only two stages within one time step. This time integration method is implemented in our multimoment global shallow-water model to build and validate a new and more efficient numerical integration framework for dynamical cores. As the key task, the numerical formulation for evaluating the derivatives in time has been developed through the Cauchy–Kowalewski procedure and the spatial discretization of the multimoment finite-volume method, which ensures fourth-order accuracy in both time and space. Several major benchmark tests are used to verify the proposed numerical framework in comparison with the existing four-stage fourth-order Runge–Kutta method, which is based on the method of lines framework. The two-stage fourth-order scheme saves about 30% of the computational cost in comparison with the four-stage Runge–Kutta scheme for global advection and shallow-water models. The proposed two-stage fourth-order framework offers a new option to develop high-performance time marching strategy of practical significance in dynamical cores for atmospheric and oceanic models.

A Numerical Shallow Water Model Based on the Non-Orthogonal Curvilinear Grids

2006 ◽  
Author(s):  
Chaofeng Tong ◽  
Yanqiu Meng
Keyword(s):  
Shallow Water ◽  
Curvilinear Coordinates ◽  
Water Model ◽  
Staggered Grid ◽  
Shallow Water Model ◽  
Governing Equations ◽  
Cartesian Coordinates ◽  
Independent Variables ◽  
Discrete Equations ◽  
Curvilinear Grids

According to the transformation relationships between the Cartesian coordinates and the general curvilinear coordinates, the governing equations of the model are derived as the forms in the general curvilinear coordinates from those in the Cartesian coordinates. In the model, the contravariant velocities are adopted as the independent variables in non-orthogonal grids. The momentum equations keep strongly conservative forms and the boundary conditions can be given easily. The model used a staggered grid arrangement. The discrete equations are solved using the SIMPLIC algorithms. The numerical model has been validated against the bifurcated flow of which the diversion angle is 30 degree. Compared with the measured values, the numerical shallow water model is shown to be capable of simulating the water domains with irregular boundaries.

scholarly journals Congested shallow water model: roof modeling in free surface flow

2018 ◽  
Vol 52 (5) ◽  
pp. 1679-1707 ◽  
Author(s):  
Edwige Godlewski ◽  
Martin Parisot ◽  
Jacques Sainte-Marie ◽  
Fabien Wahl
Keyword(s):  
Shallow Water ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Hyperbolic Equations ◽  
Mechanical Energy ◽  
Free Surface Flow ◽  
Surface Flow ◽  
Water Model ◽  
Shallow Water Model ◽  
Time Step

We are interested in the modeling and the numerical approximation of flows in the presence of a roof, for example flows in sewers or under an ice floe. A shallow water model with a supplementary congestion constraint describing the roof is derived from the Navier-Stokes equations. The congestion constraint is a challenging problem for the numerical resolution of hyperbolic equations. To overcome this difficulty, we follow a pseudo-compressibility relaxation approach. Eventually, a numerical scheme based on a finite volume method is proposed. The well-balanced property and the dissipation of the mechanical energy, acting as a mathematical entropy, are ensured under a non-restrictive condition on the time step in spite of the large celerity of the potential waves in the congested areas. Simulations in one dimension for transcritical steady flow are carried out and numerical solutions are compared to several analytical (stationary and non-stationary) solutions for validation.

Comparing Numerical Accuracy of Icosahedral A-Grid and C-Grid Schemes in Solving the Shallow-Water Model

2020 ◽  
Vol 148 (10) ◽  
pp. 4009-4033
Author(s):  
Yonggang G. Yu ◽  
Ning Wang ◽  
Jacques Middlecoff ◽  
Pedro S. Peixoto ◽  
Mark W. Govett
Keyword(s):  
Energy Conservation ◽  
Shallow Water ◽  
Time Integration ◽  
Grid Method ◽  
Maximum Error ◽  
Water Model ◽  
Shallow Water Model ◽  
Numerical Accuracy ◽  
Specific Test ◽  
Computational Performance

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

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