L-Galerkin spline methods and cut cells J. Steppeler, CSC, Hamburg Acknowledgement: Andreas Dobler (Univ. Berlin), Sanghun Park (NCAR), Edgar Huckert (SCHlA), Marc Taylor (Sandia) Joe Klemp (NCAR)

Plan of Lecture Polygonal Grids What are spline methods (Spectral methods, finite elements, Galerkin spectral elements, L-Galerkin schemes) What can do spline methods do for the modeller: Polygonal cells, High order, conservation, save computer time by sparse grids e.a, irregular resolution, cut cells Uniform approximation order, efficiency with sparse grids Examples of alternative L-Galerkin schemes C-grid scheme as low order L-Galerkin scheme What are cut cells

O2O3 and O3O3 L-Galerkin schemes O3O3 is a family of generalized L-Galerkin procedures, where nodes within a cell an be chosen at will, in particular as a regular grid. For the choice of a Gauss Lobatto Grid O3O3 is the quadrature approximated Spectral element. O2O3 (Pre-regularization) is a family of L-Galerkin Procedures on quadratic polynomial spaces, where the difference equs at even points can be chosen at will. 3 rd Order is achieved by superconvergence also on irregular grids. O2O3 and O3O3 have the standard first order conservation properties of finite elements. O3O3 has the same mimetic properties as quadrature approximated spectral elements, which it generalizes.

Sparse grids using diagnostic Points Hexagonal Grids: o2

Sparse Grids / Serendipity Order 1 Order 2 Order 3

High order finite difference scheme on the icosahedron The classical 4 th order scheme adapted to irregular grid is non-vonserving The o2o2 L-Galerkin scheme is for even i the same and the difference equs at odd points are chosen such that the L-Galerskin scheme is conserving Icosahedral grid solid body rotation advection round the earth

The L-Galerkin spectral elements SE: Standard spectral elements (Quadrature G-Lobatto points or averaging at discontinuities) L-Galerkin schemes are local and use basis functions: Pre- regularisation (simple o3), Mass conserving interpolation and more Example: C: Collocation; R: Regularization O2-O3 superconvergence is possible L-Galerkin procedure

Linear and high order basis functions x Basis functions of order 1, 2, 3 General form of L-Galerkin scheme: K: Kollocation G: Local Galerkin R: regularisaion Etc.

Simple O3: linear analysis, L-Galerkin Eq spaced L-Gal, Diffusion o4Eq spaced SE, no diffusion Eq spaced SE, Diffusion o4GL spaced SE no diffusion The 2-dx wave is 0-space for all 4 CG schemes analysed

Pre-Regularization o3-o3 equally spaced method stable, CFL with RK4: 3.9 Comparison: Standard o1 GL/SE: LA=2.7;normal Galerkin, O3 based: LA=2. conservative eq spaced:LA=2.97; eq. spaced/quadrature: unstable The o2-o3 pre-reg scheme avoids the extended 0 space

O2-O3 pre-reg scheme: 1-d Velocity u at different times irregular res for i=4,5,6,7; 1:2 change in resolution CfL = 1.8 ; cfl / 2.8 =.64 U at point 7 (x=5) for 1-d g-wave as function of time U(i=7) as function of time

O3o3-reg-nodes in cells O3o3 reg-resolution: o2 part is not determined by node points, but by a special Galerkin operation. This is the reason that we are not limited to Gauss Lobatto points. Choosing GL points results in standard quadrature Approximation. Using inner points as nodes for o3-part: CFL = 1.6 (with RK4) : diagnostic points Using outer diagnostic points for o3 part: CFL =

Standard O3 L-Galerkin O3O3 (equivalent to point quadrature with GLb) is suitable for irregular resolution

What is the o2o3 pre reg scheme: I even i uneven even i: uneven i: pre-reg- procedure i even: any finite differnce equ of at least 3 rd order This is valid for irregular resolution

O2o3 pre regularization: 2nd degree polynomials in cells: superconvergent to 3rd order initial (Boundary cells have half size of * middle cell) Time, 6 RK4 steps for cycle

Is the efficiency of sparse grids real? The dynamics part shows considerable increases in efficiency for dynamics on square cells: up to 4 : 8 for o2-o3 pre-reg, 7 : 27 for o3-o3 standard L- Galerkin, for hexagons more. For physics the savings are 1 : 8 and 1 : 27 Models of increasing complexity need to be used to investigate, if this is real. This means that the sparse grid models have the same high resolution behaviour as the full grid models.

Sparse rhomboidal-conserving serendipity elements Williamsson Test case 6 Initial Day1Day 10

o3o3 standard scheme with GL points: Third (fourth) Order Convergence of Shallow Water Model at Day 3

fu Hexagons o2o3 Pre-Reg t2t2 t1t1 t3t3 Infinitesimal Control volume e1e1 e2e2 e3e3 Baumgardner Cloud for 2nd Ord differnce Full grid Directional Derivative: 1,x,y,x 2,y 2 xy 7 point hex Main node Stencil

The 10 pnt 3rd order stencil for the edge point Dx= AUse the cloud:1,2,3,4,5,6,7,8,9,10 to compute the derivative along the edges at 0 and analogue on all centres of edges (3,5) BUse this to create 3 rd order polynomials on all edges CUse To compute derivative at 1.

The hexagonal grid

Hexagon sparse grid O2 / o3 (pre- regularization) advection in x-direction

Cut cells in a high (2 nd) order grid Only a part of the cell is taken by the atmosphere

Basic Facts on Cut Cells Mountains must be represented by linear of higher order splines, not piecewise constants. A (false) piecewise constant mountain representation can lead to lack of convergence (Gallus and Klemp (2000), MWR 128,1153). A linear spline cut cell mountain representation handles the smooth shallow mountain as the terrain following co-ordinate. (Steppeler et al. (2002),MWR 130, 2143) It is much better for steep mountains (even vertical). This leads to improved short range forecasts and improved longer range forecasts, also to an improved precipitation climatology. (Steppeler et al. (2006),MWR 134, 3625; (2011) Atm. Sci. Let. 12, 340;(2013) Geosci. Model Dev., 6, )

Cut cells: Monthly precip in 5 d Forecasts Precip: a, control, b, obs, c, cut cell, d, CLM

3 rd order spectral elements with cut cells: spectral elements of order 3 (Pre-reg o2o3):

Cut cells, curved circular bounday o2o3 3rd order L-Galerkin scheme Cell combination; red: inner Boundary amplitudes; blue: potential outer boundary points Cell combination means that blue points are done by extrapolation

Cut cells, curved circular bounday o2o3 Boundary amplitudes are forecasted and then changed according to boundary conditions: Cell combination means that some of the outer boundary points (blue) are extrapolated rather than forecasted Here the simplest case is used, where all blue points are diagnostic. This means a loss of resolution at the boundary. Boundary integral for field f i,j associated with node I,j:

Open circular boundary

Basis functions for C-grid Scheme u h U: lin in z, const in x W: const in z, lin in x Density and T: lin in z and x

The Toolbox to make Cut Cells working Prefer linear or higher order interpolation over piecewise constant Diffusion Vertical Implicit “Thin Wall Approximation” Cell Combination for larger CFL (horizontal or vertical) Cell combination will also fight noise problems For LMZ: compute advection in undisturbed cells and copy to boundary cells

Increased diffusion in lowest laye Control Diff.001 Diff..05 for k=1,2

Cut cells, cell combination for U-field only Piecew const splines Piecew linear splines

Impact of cut cells on cyclonic forecasts in medium range Cut cells 10 days, u10m Observation Terrain following, 10 days, u10m For tropical forecasts see: Atmos. Sci. lett. (2011), pp; Geo. Mod. dev. (2013) pp

Questions? Thank you for listening

Cut Cells, Triangulation (irregular), high order points, phantom points, diagnostic points

Characteristic polynomials

B Numerical properties of different Serendipity Schemes

Cut cells in high (3 rd ) oder Organ pipe with lowest cell cut in half:

Alternative L-Galerkin methods on CG function spaces on cells o3o3 standard L-Galerkin: is stable and conserving for any choice of grid spacing, in particular equally spaced grids. For the choice of Gauss Lobatto points it is identical to spectral elements with quadrature approximation. o2o3 pre-regularization: Achieves (at least) 3 rd order approximation for quadratic function spaces. Linear conservation is present for all L-Galerkin methods. Standard l- Gaerkin has been shown to have further mimetic properties.