Simulation of Shallow Water Jets with a Unified Element-based Continuous/Discontinuous Galerkin Model with Grid Flexibility on the Sphere

We test the behaviour of a unified continuous/discontinuous Galerkin (CG/DG) shallow-water model in spherical geometry with curved elements on three different grids of ubiquitous use in atmospheric modelling: (i) the cubed-sphere, (ii) the reduced latitude–longitude, and (iii) the icosahedral grid. Both conforming and non-conforming grids are adopted including static and dynamically adaptive grids for a total of twelve mesh configurations. The behaviour of CG and DG on the different grids are compared for a nonlinear midlatitude perturbed jet and for a linear case that admits an analytic solution. Because the inviscid solution on certain grids shows a high sensitivity to the resolution, the viscous counterpart of the governing equations is also solved and the results compared. The logically unstructured element-based CG/DG model described in this article is flexible with respect to arbitrary grids. However, we were unable to define a best grid configuration that could possibly minimize the error regardless of the characteristic geometry of the flow. This is especially true if the governing equations are not regularized by the addition of a sufficiently large, fully artificial, diffusion mechanism, as will be shown. The main novelty of this study lies in the unified implementation of two element-based Galerkin methods that share the same numerical machinery and do not rely on any specific grid configuration to solve the shallow-water equation on the sphere.

[1]  Francis X. Giraldo,et al.  Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling , 2015 .

[2]  Paul A. Ullrich,et al.  The spectral element method (SEM) on variable-resolution grids: evaluating grid sensitivity and resolution-aware numerical viscosity , 2014 .

[3]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[4]  R. K. Scott,et al.  An initial-value problem for testing numerical models of the global shallow-water equations , 2004 .

[5]  J. G. Verwer,et al.  RAPPORT Spatial Discretization of the Shallow Water Equations in Spherical Geometry using Osher ’ s Scheme , 1999 .

[6]  W J Gordon,et al.  CONSTRUCTION OF CURVILINEAR COORDINATE SYSTEMS AND APPLICATION AND APPROXIMATION TO MESH GENERATION , 1973 .

[7]  Francis X. Giraldo,et al.  Dry and moist idealized experiments with a two-dimensional spectral element model , 2012 .

[8]  Ronald Gelaro,et al.  The NOGAPS Forecast Model: A Technical Description , 1991 .

[9]  G. Starius,et al.  Composite mesh difference methods for elliptic boundary value problems , 1977 .

[10]  M. Taylor.  The Spectral Element Method for the Shallow Water Equations on the Sphere , 1997 .

[11]  Francis X. Giraldo,et al.  An LES-like stabilization of the spectral element solution of the euler equations for atmospheric flows , 2014 .

[12]  J. Wolf,et al.  The scaled boundary finite-element method – alias consistent infinitesimal finite element cell method – for diffusion , 1999 .

[13]  R. Courant,et al.  On the Partial Difference Equations, of Mathematical Physics , 2015 .

[14]  J. Côté,et al.  A Lagrange multiplier approach for the metric terms of semi‐Lagrangian models on the sphere , 1988 .

[15]  Saulo R. M. Barros,et al.  Analysis of grid imprinting on geodesic spherical icosahedral grids , 2013, J. Comput. Phys..

[16]  J. F. Thompson,et al.  A general three-dimensional elliptic grid generation system on a composite block structure , 1987 .

[17]  Francis X. Giraldo,et al.  Comparison between adaptive and uniform discontinuous Galerkin simulations in dry 2D bubble experiments , 2013, J. Comput. Phys..

[18]  P. Paolucci,et al.  The “Cubed Sphere” , 1996 .

[19]  Akio Arakawa,et al.  Integration of the Nondivergent Barotropic Vorticity Equation with AN Icosahedral-Hexagonal Grid for the SPHERE1 , 1968 .

[20]  R. Sadourny.  Conservative Finite-Difference Approximations of the Primitive Equations on Quasi-Uniform Spherical Grids , 1972 .

[21]  N. A. Phillips,et al.  A Map Projection System Suitable for Large-scale Numerical Weather Prediction , 1957 .

[22]  Steven J. Ruuth,et al.  A New Class of Optimal High-Order Strong-Stability-Preserving Time Discretization Methods , 2002, SIAM J. Numer. Anal..

[23]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[24]  J. Boyd,et al.  The Erf1⁄2-Log Filter and the Asymptotics of the Euler and Vandeven Sequence Accelerations , 1999 .

[25]  Lei Bao,et al.  A mass and momentum flux-form high-order discontinuous Galerkin shallow water model on the cubed-sphere , 2014, J. Comput. Phys..

[26]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[27]  John M. Dennis,et al.  A Comparison of Two Shallow-Water Models with Nonconforming Adaptive Grids , 2008 .

[28]  Emil M. Constantinescu,et al.  Implicit-Explicit Formulations of a Three-Dimensional Nonhydrostatic Unified Model of the Atmosphere (NUMA) , 2013, SIAM J. Sci. Comput..

[29]  Jan S. Hesthaven,et al.  Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations , 2002 .

[30]  Francis X. Giraldo,et al.  A spectral element shallow water model on spherical geodesic grids , 2001 .

[31]  Francis X. Giraldo,et al.  Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode , 2012, J. Comput. Phys..

[32]  Stephen J. Thomas,et al.  A Discontinuous Galerkin Global Shallow Water Model , 2005, Monthly Weather Review.

[33]  Francis X. Giraldo,et al.  Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations , 2014, J. Comput. Phys..

[34]  G. L. Browning,et al.  A comparison of three numerical methods for solving differential equations on the sphere , 1989 .

[35]  Aimé Fournier,et al.  Voronoi, Delaunay, and Block-Structured Mesh Refinement for Solution of the Shallow-Water Equations on the Sphere , 2009 .

[36]  Jean Côté,et al.  Experiments with different discretizations for the shallow‐water equations on a sphere , 2012 .