1. A COMPARISON OF SOME SHALLOW WATER TEST CASES IN HOMME USING NUMERICALLY AND ANALYTICALLY COMPUTED TRANSFORMATIONS Numerical Approximations of Coordinate Transformations Dan Whitt Summer 2009 SIParCS Internship National Center for Atmospheric Research Mentor: Amik St-Cyr

2. HOMME – Cubed Sphere Geometry Image 1 available at: http://141.104.22.210/Div/Winchester/jhhs/math/probweek/p2004/a022304.html Image 2 available at: http://www.csc.cs.colorado.edu/resources/research_images/homme/cubed_sphere.jpg

3. These pictures are from Ram Nair’s website. A Central Gnomonic Projection from the cube to the surface of the sphere.

4. Curvilinear Coordinates Physical Domain: S 2, the sphere, broken up into 6 equivalent regions. Computational Domains: 6 copies of [-1,1] x [-1,1] in R 2. (one for each region in the physical domain) Objects of Interest: Scalar and vector fields on S 2. Current Physical Basis: the local λ and θ basis vectors, which represent orthogonal unit vectors in the longitude and latitude direction respectively. E.g.: v = v 1 λ + v 2 θ Alternative Physical Basis: the global X, Y, Z, Cartesian basis. E.g. v = v 1 X + v 2 Y +v 3 Z Computational Basis: usual x,y Cartesian basis in R 2.

5. Transforming from the Sphere to the Cube The projection of the local latitude/longitude coordinate system is non-orthogonal. Hence, the projected vector admits covariant and contravariant representations. (And vice versa) Figure available at: http://www.cgd.ucar.edu/research/abstracts/images/2007/gnomonic.jpg

6. Contra-variant components Covariant components Contra-variant basis vectors Covariant basis vectors The vector X Contravariant and Covariant Vector Representations

7. Defining the Transformation Use local transformation matrices, D, D -1, and covariant and contra-variant metric tensors, g jk, g jk.

8. More Precisely… - D transforms contra- variant components on the cube face to the physical domain on the sphere. - The covariant metric tensor is the inverse of the contra-variant metric tensor. OR Two Observations: Definition of D, the coordinate transformation matrix:

9. Current HOMME Procedure We analytically differentiate the transformation and use those formulas to obtain the transformation terms at each point on the mesh. Easy to do, but one cannot have a mesh point on the pole.

10. Problem The Discrete Formulation of the Metric Identities is satisfied if and only if the interpolant (I N ) of the metric terms is divergence free. If we do not satisfy the metric identities, we introduce errors.

11. Conservation and Metric Identities If we initialize HOMME with a steady state problem, like SWTC2, a steady-state flow over the globe, we expect to find that the solution remains constant in time. Since the flux is constant in space, its divergence must be zero, and the solution must remain constant. This is the concept of “free-stream preservation” in CFD, which requires that a uniform flow remain uniform in time. (Why we care)

12. How do we fix this? Find a useful Theorem. The discrete metric identities will be satisfied if we approximate the coordinate transformation terms to the same order N as we approximate the solution. See: Kopriva, D.A. Metric Identities and the Discontinuous Spectral Element Method. Journal Sci. Comput. Vol. 26. No. 3. March 2006.

13. Numerical Approximation We approximate the coordinate transformation using the same order finite difference operator we use to obtain the solution.

14. How we do that? 1. Partition the computational domain. Let the set N of points be the GLL quadrature nodes on the set [-1,1]. 2. Take the interpolation projections of the transformation functions X(ξ,η), Y(ξ,η), and Z(ξ,η), where the {π i (ξ)} i=1,…,N are the basis of orthogonal polynomials. e.g.: 3. Differentiate to obtain the approximate values at the nodes by applying the following differentiation matrix: (A 3D-Cartesian example)

15. My Work We numerically compute all the coordinate transformation terms and compare results with original HOMME runs to see the effects. Utilizing 3D-Cartesian space also allows us to run HOMME with any number of elements, eliminating the problem of having a point on the pole.

16. Not Finished Yet