scholarly journals Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Thida Pongsanguansin ◽  
Montri Maleewong ◽  
Khamron Mekchay
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Weighted Average ◽  
Flat Bottom ◽  
Flux Function ◽  
Dg Method ◽  
Average Flux ◽  
Steady Solutions

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

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.

scholarly journals Modified Predictor-Corrector WAF Method for the Shallow Water Equations with Source Terms

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Montri Maleewong
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Numerical Solutions ◽  
Piecewise Linear ◽  
Weighted Average ◽  
Test Cases ◽  
Half Time ◽  
Source Terms ◽  
Time Step ◽  
Predictor Corrector

A modified predictor-corrector scheme combining with the depth gradient method (DGM) and the weighted average flux (WAF) method has been presented to solve the one-dimensional shallow water equations with source terms. Approximate solutions in the predictor step are obtained by the DGM with piecewise-linear reconstructions in each cell volume. The source terms can then be calculated directly by these predicted values at the corresponding half-time step. In the corrector step, the TVD version of the WAF method is applied to calculate the numerical fluxes at the same half-time step for each cell face. The accuracy of numerical solutions is shown by applying the method to solve various test cases in both steady and unsteady problems with and without source terms. It shows that the numerical results are in good agreement with the existing analytical solutions as well as experimental data in some test cases.

scholarly journals A Discontinuous Galerkin Transport Scheme on the Cubed Sphere

2005 ◽  
Vol 133 (4) ◽  
pp. 814-828 ◽  
Author(s):  
Ramachandran D. Nair ◽  
Stephen J. Thomas ◽  
Richard D. Loft
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Spherical Surface ◽  
Time Integration ◽  
Spectral Element ◽  
Projection Methods ◽  
Central Projection ◽  
Error Metrics ◽  
Dg Method ◽  
Cubed Sphere

A conservative transport scheme based on the discontinuous Galerkin (DG) method has been developed for the cubed sphere. Two different central projection methods, equidistant and equiangular, are employed for mapping between the inscribed cube and the sphere. These mappings divide the spherical surface into six identical subdomains, and the resulting grid is free from singularities. Two standard advection tests, solid-body rotation and deformational flow, were performed to evaluate the DG scheme. Time integration relies on a third-order total variation diminishing (TVD) Runge–Kutta scheme without a limiter. The numerical solutions are accurate and neither exhibit shocks nor discontinuities at cube-face edges and vertices. The numerical results are either comparable or better than a standard spectral element method. In particular, it was found that the standard relative error metrics are significantly smaller for the equiangular as opposed to the equidistant projection.

Quadrature-Free Implementation of a Discontinuous Galerkin Global Shallow-Water Model via Flux Correction Procedure

2015 ◽  
Vol 143 (4) ◽  
pp. 1335-1346 ◽  
Author(s):  
Ramachandran D. Nair
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Water Model ◽  
Free Form ◽  
Alternative Formulation ◽  
Test Case ◽  
Computationally Efficient ◽  
Weak Formulation ◽  
Hyperbolic Conservation Law ◽  
Dg Method

Abstract The discontinuous Galerkin (DG) discretization relies on an integral (weak) formulation of the hyperbolic conservation law, which leads to the evaluation of several surface and line integrals for multidimensional problems. An alternative formulation of the DG method is possible under the flux reconstruction (FR) framework, where the equations are solved in differential form and the discretization is free from quadrature rules, resulting in computationally efficient algorithms. The author has implemented a quadrature-free form of the nodal DG method based on the FR approach combined with spectral differencing (SD), in a shallow-water (SW) model employing cubed-sphere geometry. The performance of the SD model is compared with the regular nodal DG variant of the SW model using several benchmark tests, including a viscous test case. A positivity-preserving local filter is tested for SD advection that removes spurious oscillations while being conservative and accurate. In this implementation, the SD formulation is found to be 18% faster than the DG method for inviscid SW tests cases and 24% faster for the viscous case. The results obtained by the SD formulation are on par with the regular nodal DG formulation in terms of accuracy and convergence.

A Three-Dimensional Parallel Discontinuous Galerkin Solver for Acoustic Propagation Studies

2003 ◽  
Vol 2 (2) ◽  
pp. 157-173 ◽  
Author(s):  
A. Crivellini ◽  
F. Bassi
Keyword(s):  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Acoustic Propagation ◽  
High Order ◽  
Runge Kutta Method ◽  
Recent Developments ◽  
Dg Method ◽  
Hybrid Grids ◽  
Linux Cluster

The paper presents recent developments of a computational code for the numerical investigation of acoustic propagation. The code solves the three-dimensional linear Euler equations using a Discontinuous Galerkin (DG) method for the spatial discretization and an explicit high-order low-storage Runge-Kutta method for advancing the solution in time. Thanks to DG discretization, high-order accurate numerical solutions on arbitrary unstructured hybrid grids have been easily computed. The code has been parallelized using MPI and preliminary results on a small 10-processor Linux cluster seem very promising.

A Weighted Average Flux (WAF) scheme applied to shallow water equations for real-life applications

2013 ◽  
Vol 62 ◽  
pp. 155-172 ◽  
Author(s):  
Riadh Ata ◽  
Sara Pavan ◽  
Sofiane Khelladi ◽  
Eleuterio F. Toro
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Real Life ◽  
Weighted Average ◽  
Average Flux
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