A Nonoscillatory Discontinuous Galerkin Transport Scheme on the Cubed Sphere

2012 ◽  
Vol 140 (9) ◽  
pp. 3106-3126 ◽  
Author(s):  
Yifan Zhang ◽  
Ramachandran D. Nair
Keyword(s):  
Numerical Solution ◽  
Discontinuous Galerkin ◽  
Curvilinear Coordinates ◽  
Tracer Transport ◽  
Third Order ◽  
Positivity Preserving ◽  
Weighted Essentially Nonoscillatory ◽  
Sphere Geometry ◽  
Cubed Sphere ◽  
Atmospheric Tracer

Abstract The discontinuous Galerkin (DG) method is high order, conservative, and offers excellent parallel efficiency. However, when there are discontinuities in the solution, the DG transport scheme generates spurious oscillations that reduce the quality of the numerical solution. For applications such as the atmospheric tracer transport modeling, a nonoscillatory, positivity-preserving solution is a basic requirement. To suppress the oscillations in the DG solution, a limiter based on the Hermite-Weighted Essentially Nonoscillatory (H-WENO) method has been implemented for a third-order DG transport scheme. However, the H-WENO limiter can still produce wiggles with small amplitudes in the solutions, but this issue has been addressed by combining the limiter with a bound-preserving (BP) filter. The BP filter is local and easy to implement and can be used for making the solution strictly positivity preserving. The DG scheme combined with the limiter and filter preserves the accuracy of the numerical solution in the smooth regions while effectively eliminating overshoots and undershoots. The resulting nonoscillatory DG scheme is third-order accurate (P2-DG) and based on the modal discretization. The 2D Cartesian scheme is further extended to the cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates. The accuracy and effectiveness of the limiter and filter are demonstrated with several benchmark tests on both the Cartesian and spherical geometries.

scholarly journals Two conservative multi-tracer efficient semi-Lagrangian schemes for multiple processor systems integrated in a spectral element (climate) dynamical core

2016 ◽  
Vol 7 (3) ◽  
pp. 74-98 ◽  
Author(s):  
Christoph Erath ◽  
Mark A. Taylor ◽  
Ramachandran D. Nair
Keyword(s):  
Spectral Element Method ◽  
Spectral Element ◽  
Tracer Transport ◽  
Third Order ◽  
Dynamical Core ◽  
Community Atmosphere Model ◽  
High Scalability ◽  
Passive Tracers ◽  
Cubed Sphere ◽  
A Current

Abstract In today’s atmospheric numerical modeling, scalable and highly accurate numerical schemes are of particular interest. To address these issues Galerkin schemes, such as the spectral element method, have received more attention in the last decade. They also provide other state-of-the-art capabilities such as improved conservation. However, the tracer transport of hundreds of tracers, e.g., in the chemistry version of the Community Atmosphere Model, is still a performance bottleneck. Therefore, we consider two conservative semi-Lagrangian schemes. Both are designed to be multi-tracer efficient, third order accurate, and allow significantly longer time steps than explicit Eulerian formulations. We address the difficulties arising on the cubed-sphere projection and on parallel computers and show the high scalability of our approach. Additionally, we use the two schemes for the transport of passive tracers in a dynamical core and compare our results with a current spectral element tracer transport advection used by the High-Order Method Modeling Environment.

scholarly journals A Positivity-preserving Conservative Semi-Lagrangian Multi-moment Global Transport Model on the Cubed Sphere

2021 ◽  
Vol 38 (9) ◽  
pp. 1460-1473 ◽  
Author(s):  
Jie Tang ◽  
Chungang Chen ◽  
Xueshun Shen ◽  
Feng Xiao ◽  
Xingliang Li
Keyword(s):  
Single Cell ◽  
Rational Function ◽  
Finite Volume ◽  
Transport Model ◽  
Practical Implementation ◽  
Finite Volume Scheme ◽  
Courant Number ◽  
Positivity Preserving ◽  
Sphere Geometry ◽  
Cubed Sphere

AbstractA positivity-preserving conservative semi-Lagrangian transport model by multi-moment finite volume method has been developed on the cubed-sphere grid. Two kinds of moments (i.e., point values (PV moment) at cell interfaces and volume integrated average (VIA moment) value) are defined within a single cell. The PV moment is updated by a conventional semi-Lagrangian method, while the VIA moment is cast by the flux form formulation to assure the exact numerical conservation. Different from the spatial approximation used in the CSL2 (conservative semi-Lagrangian scheme with second order polynomial function) scheme, a monotonic rational function which can effectively remove non-physical oscillations is reconstructed within a single cell by the PV moments and VIA moment. To achieve exactly positive-definite preserving, two kinds of corrections are made on the original conservative semi-Lagrangian with rational function (CSLR) scheme. The resulting scheme is inherently conservative, non-negative, and allows a Courant number larger than one. Moreover, the spatial reconstruction can be performed within a single cell, which is very efficient and economical for practical implementation. In addition, a dimension-splitting approach coupled with multi-moment finite volume scheme is adopted on cubed-sphere geometry, which benefitsthe implementation of the 1D CSLR solver with large Courant number. The proposed model is evaluated by several widely used benchmark tests on cubed-sphere geometry. Numerical results show that the proposed transport model can effectively remove nonphysical oscillations and preserve the numerical non-negativity, and it has the potential to transport the tracers accurately in a real atmospheric model.

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.

A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere

2014 ◽  
Vol 142 (1) ◽  
pp. 457-475 ◽  
Author(s):  
Wei Guo ◽  
Ramachandran D. Nair ◽  
Jing-Mei Qiu
Keyword(s):  
Discontinuous Galerkin ◽  
Strong Stability ◽  
Numerical Approach ◽  
Atmospheric Modeling ◽  
Curvilinear Coordinates ◽  
Major Drawback ◽  
Time Stepping ◽  
Wide Range ◽  
Cubed Sphere ◽  
Dg Methods

Abstract The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.

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.

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