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.

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 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 Asynchronous Communication in Spectral Element and Discontinuous Galerkin Methods for Atmospheric Dynamics

2016 ◽  
Author(s):  
Benjamin F. Jamroz ◽  
Robert Klöfkorn
Keyword(s):  
Discontinuous Galerkin ◽  
Large Scale ◽  
Discontinuous Galerkin Methods ◽  
Time Integration ◽  
Spectral Element ◽  
Asynchronous Communication ◽  
Galerkin Methods ◽  
Data Movement ◽  
Significant Performance ◽  
The Cost

Abstract. The scalability of computational applications on current and next generation supercomputers is increasingly limited by the cost of inter-process communication. We implement non-blocking asynchronous communication in the High-Order Methods Modeling Environment for the time-integration of the hydrostatic fluid equations using both the Spectral Element and Discontinuous Galerkin methods. This allows the overlap of computation with communication effectively hiding some of the costs of communication. A novel detail about our approach is that it provides some data movement to be performed during the asynchronous communication even in the absence of other computations. This method produces significant performance and scalability gains in large-scale simulations.

scholarly journals Implementation of high-order, discontinuous Galerkin time stepping for fractional diffusion problems

2021 ◽  
Vol 62 ◽  
pp. 121-147
Author(s):  
William McLean
Keyword(s):  
Discontinuous Galerkin ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Time Integration ◽  
Fractional Diffusion ◽  
Practical Implementation ◽  
Time Stepping ◽  
Dg Method ◽  
Carry Over ◽  
Diffusion Problems

The discontinuous Galerkin (DG) method provides a robust and flexible technique for the time integration of fractional diffusion problems. However, a practical implementation uses coefficients defined by integrals that are not easily evaluated. We describe specialized quadrature techniques that efficiently maintain the overall accuracy of the DG method. In addition, we observe in numerical experiments that known superconvergence properties of DG time stepping for classical diffusion problems carry over in a modified form to the fractional-order setting. doi: 10.1017/S1446181120000152

Time-Domain Numerical Solutions of Maxwell Interface Problems with Discontinuous Electromagnetic Waves

2016 ◽  
Vol 8 (3) ◽  
pp. 353-385 ◽  
Author(s):  
Ya Zhang ◽  
Duc Duy Nguyen ◽  
Kewei Du ◽  
Jin Xu ◽  
Shan Zhao
Keyword(s):  
Discontinuous Galerkin ◽  
Time Domain ◽  
Electromagnetic Waves ◽  
Numerical Solutions ◽  
Time Integration ◽  
Transverse Magnetic ◽  
Interface Problems ◽  
Jump Conditions ◽  
Material Interface ◽  
Wave Solutions

Abstract.This paper is devoted to time domain numerical solutions of two-dimensional (2D) material interface problems governed by the transverse magnetic (TM) and transverse electric (TE) Maxwell's equations with discontinuous electromagnetic solutions. Due to the discontinuity in wave solutions across the interface, the usual numerical methods will converge slowly or even fail to converge. This calls for the development of advanced interface treatments for popular Maxwell solvers. We will investigate such interface treatments by considering two typical Maxwell solvers – one based on collocation formulation and another based on Galerkin formulation. To restore the accuracy reduction of the collocation finite-difference time-domain (FDTD) algorithm near an interface, the physical jump conditions relating discontinuous wave solutions on both sides of the interface must be rigorously enforced. For this purpose, a novel matched interface and boundary (MIB) scheme is proposed in this work, in which new jump conditions are derived so that the discontinuous and staggered features of electric and magnetic field components can be accommodated. The resulting MIB time-domain (MIBTD) scheme satisfies the jump conditions locally and suppresses the staircase approximation errors completely over the Yee lattices. In the discontinuous Galerkin time-domain (DGTD) algorithm – a popular GalerkinMaxwell solver, a proper numerical flux can be designed to accurately capture the jumps in the electromagnetic waves across the interface and automatically preserves the discontinuity in the explicit time integration. The DGTD solution to Maxwell interface problems is explored in this work, by considering a nodal based high order discontinuous Galerkin method. In benchmark TM and TE tests with analytical solutions, both MIBTD and DGTD schemes achieve the second order of accuracy in solving circular interfaces. In comparison, the numerical convergence of the MIBTD method is slightly more uniform, while the DGTD method is more flexible and robust.

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.

scholarly journals A Hybrid Time Integration Scheme for the Discontinuous Galerkin Discretizations of Convection-Dominated Problems

Energies ◽  
2020 ◽  
Vol 13 (8) ◽  
pp. 1870
Author(s):  
Liang Li ◽  
Songping Wu
Keyword(s):  
Discontinuous Galerkin ◽  
Time Integration ◽  
Integration Scheme ◽  
Hybrid Scheme ◽  
Time Step ◽  
Third Order ◽  
Element Size ◽  
Time Integration Scheme ◽  
Dg Method ◽  
Convection Dominated

Discontinuous Galerkin (DG) method is a popular high-order accurate method for solving unsteady convection-dominated problems. After spatially discretizing the problem with the DG method, a time integration scheme is necessary for evolving the result. Owing to the stability-based restriction, the time step for an explicit scheme is limited by the smallest element size within the mesh, making the calculation inefficient. In this paper, a hybrid scheme comprising a three-stage, third-order accurate, and strong stability preserving Runge–Kutta (SSP-RK3) scheme and the three-stage, third-order accurate, L-stable, and diagonally implicit Runge–Kutta (LDIRK3) scheme is proposed. By dealing with the coarse and the refined elements with the explicit and implicit schemes, respectively, the time step for the hybrid scheme is free from the limitation of the smallest element size, making the simulation much more efficient. Numerical tests and comparison studies were made to show the performance of the hybrid scheme.

scholarly journals Development and evaluation of a hydrostatic dynamical core using the spectral element/discontinuous Galerkin methods

2014 ◽  
Vol 7 (3) ◽  
pp. 4119-4151
Author(s):  
S.-J. Choi ◽  
F. X. Giraldo
Keyword(s):  
Discontinuous Galerkin ◽  
Radial Direction ◽  
Numerical Solutions ◽  
Baroclinic Instability ◽  
Three Dimensional ◽  
Spectral Element ◽  
Initial State ◽  
Dynamical Core ◽  
Lagrange Polynomials ◽  
Cartesian Space

Abstract. In this paper, we present a dynamical core for the atmospheric primitive hydrostatic equations using a unified formulation of spectral element (SE) and discontinuous Galerkin (DG) methods in the horizontal direction with a finite difference (FD) method in the radial direction. The CG and DG horizontal discretization employs high-order nodal basis functions associated with Lagrange polynomials based on Gauss–Lobatto–Legendre (GLL) quadrature points, which define the common machinery. The atmospheric primitive hydrostatic equations are solved on the cubed-sphere grid using the flux form governing equations in a three-dimensional (3-D) Cartesian space. By using Cartesian space, we can avoid the pole singularity problem due to spherical coordinates and this also allows us to use any quadrilateral-based grid naturally. In order to consider an easy way for coupling the dynamics with existing physics packages, we use a FD in the radial direction. The models are verified by conducting conventional benchmark test cases: the Rossby–Haurwitz wavenumber 4, Jablonowski–Williamson tests for balanced initial state and baroclinic instability, and Held–Suarez tests. The results from those tests demonstrate that the present dynamical core can produce numerical solutions of good quality comparable to other models.

Semi-Lagrangian exponential time-integration method for the shallow water equations on the cubed sphere grid

2020 ◽  
Vol 35 (6) ◽  
pp. 355-366
Author(s):  
Vladimir V. Shashkin ◽  
Gordey S. Goyman
Keyword(s):  
Shallow Water ◽  
Gravity Waves ◽  
Shallow Water Equations ◽  
Time Integration ◽  
Standard Test ◽  
Integration Scheme ◽  
Test Problems ◽  
Cubed Sphere ◽  
Lagrangian Scheme ◽  
Inertia Gravity Waves

AbstractThis paper proposes the combination of matrix exponential method with the semi-Lagrangian approach for the time integration of shallow water equations on the sphere. The second order accuracy of the developed scheme is shown. Exponential semi-Lagrangian scheme in the combination with spatial approximation on the cubed-sphere grid is verified using the standard test problems for shallow water models. The developed scheme is as good as the conventional semi-implicit semi-Lagrangian scheme in accuracy of slowly varying flow component reproduction and significantly better in the reproduction of the fast inertia-gravity waves. The accuracy of inertia-gravity waves reproduction is close to that of the explicit time-integration scheme. The computational efficiency of the proposed exponential semi-Lagrangian scheme is somewhat lower than the efficiency of semi-implicit semi-Lagrangian scheme, but significantly higher than the efficiency of explicit, semi-implicit, and exponential Eulerian schemes.

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