Recent developments in Numerical Methods, incl. report from the PDEs on the sphere meeting(Part II)

Michael Baldauf (DWD)

WGNE-Meeting18.-22. Oct. 2010, Tokyo

Finite Volume schemes

advantages• 'automatically' have local conservation properties of the prognostic variables• designed to handle shocks and other discontinuities• can easily made to satisfy monotonicity and positivity constraints• standard method in many CFD fields

presentations:• Riemann solvers for 2D shallow water equations [Castro]• convergence proofs for FV methods on Riemannian manifolds [T. Müller]• higher order Riemann solvers, proper choice of the flux formulation

(AUSM+-up is superior to the Rusanov flux) [Ullrich]

Spatial discretization methods

Riemann solver for shallow water equations (Tsunami-simulation)[Castro]• ADER scheme• very accurate treatment of shocks and other discontinuities (solution of a

generalized Riemann problem)• well-balancing ('lake at rest'): exactly by numerical reconstruction of the

bathymetry• local time stepping for higher efficiency

homogeneous Riemann-problem

generalized Riemann-problem

from Castro, Toro (2008) JCP

Spatial discretization methods

from: C. Castro, talk at 'PDE's on the sphere', Aug. 2010, Potsdam

Mediterranean tsunami

Arbitrary system of PDE's:

Weak formulation of the PDE-system with test functions v(r):

Representation by base functions pl(r):

This leads to an ODE-System for the coefficients Qj,l(k)(t)

Discontinuous Galerkin Method

To choose:• flux formulation for boundary integral (Lax-Friedrich, Rusanov, AUSM+, ...)• quadrature formula (Gauss-Lobatto, ...)• time integration scheme (Runge-Kutta, implicit, ...)

Spatial discretization methods

M. Baldauf (DWD) 20.04.2010 6

Discontinuous Galerkin Method

• Seek weak solutions of a balance equation(correspondance to finite volume methods conservation)

• Expand solution into a sum of base functions on each grid cell (correspondance to finite element methods higher order)(classical Finite-volume-scheme is recovered by base function = const.)

• useable on arbitrary grids suitable for complex geometries• discontinuous elements mass matrix is block-diagonal• in combination with an explicit time integration scheme

(e.g. Runge-Kutta RKDG-methods) highly parallelizable codebut: how to solve vertically expanding sound waves efficiently?in general: max. allowable Courant numbers are quite small

Spatial discretization methods

• Application to LES [A. Müller]• hp-adaptive method for shallow water equations [Blaise]:

DG well suited for adaptive mesh refinement (AMR)• shallow-water equations on the sphere with curvilinear coordinates [Nair]

Spectral elements (SE) method = continuous Galerkin methodfor non-hydro. compressible models (global, regional)3D semi-implicit (GMRES)benefits:1.) higher order accuracy2.) flexibility on unstructured and adaptive grids3.) efficient on massively parallel computers[Kelly]

'High Order Methods Modeling Environment' (HOMME) [Levy]HOMME-spectral element method needs quadrilateral grids cubed sphererefinement

Spatial discretization methods

Finite element methods

more often used in oceanographic applications [Danilov], [Sidorenko]:horizontally highly unstructured and stretched grids

Equation systems for 3D atmospheric models

non-hydrostatic, compressible equation sets

anelastic equation sets• for global CSU-model development: [Heikes]

• anelastic Z-grid model (i.e. vertical vorticity , horizontal divergence , vertical velocity w, Exner function ')

• anelastic vector-vorticity-eqns. (Jung, Arakawa (2008) MWR)• Lipps, Hemler (1982) eqns. [Smolarkiewicz], [Szmelter]

Mesh Refinement (AMR)

Adaptive (AMR)- 'amatos' (software package (Behrend) use of space filling curves with DG discretization [A. Müller]- wavelets [Dubos]

Static:- by non-conformal refinement in triangular grid in ICON model [Zängl]- HOMME- by stretching function of hexagon grid in MPAS model [Klemp]

Grid structure (schematic view)

Triangles are used as primal cells

Mass points are in the circumcenter

Velocity is defined at the edge midpoints

Red cells refer to refined domain

Boundary interpolation is needed from parent to child mass points and velocity points

ICON (G. Zängl, DWD)

Temperature at lowest model level on day 14

70 km

70 km, nested nest, 35 km

35 km

Mesh refinement in ICON (G. Zängl, DWD)

staggered grids have: (+) advantages in the dispersion relation (high wavenumbers)

(--) disadvantages towards unstaggered grids:• stronger timestep constraint• horizontal-vertical splitting a bit easier in unstagg. grids • in the vicinity of changes in the grid resolution, much stronger wave

reflection than in unstaggered grids can occur [Ullrich], [Thuburn]important for AMR!

change of grid position in one time-step: staggered for gravity wave, unstaggered elsewhere: good dispersion relations and no Gibbs-phenomena in CCAM [McGregor]

probably the distinction between these two grids types becomes less important for higher order methods (?)

staggered vs. unstaggered grids

Linearised shallow water equations

analytic dispersion relation ( ):

'Standard' C-grid spatial discretisation:• divergence: sum of fluxes over the edges (Gauss theorem)• gradient: centered differences ( -> 2nd order )• Coriolis term: Reconstruction of v by next 4 neighbours with

RBF-vector reconstruction (with an 'idealised' RBF-fct. =1)temporal discretisation: assumed exact

inertial-gravity wave(Poincaré wave)

geostrophic mode

Rossby-deformation radius: LR = c / f

[Baldauf]

ICON-project

Triangle grid

interpretation 1:internal excitation of an elementary cell

interpretation 2:(poor) resolution of shorter waves (until 'x') ( not artificial)

1st Brillouin-zone

consequences in practice:reduction of time step

5 branches: 1,2 = fI-G(k), 3,4 = fHF(k), 5=0 (exact in equilat. grid!)

ICON-project

‚high-frequent mode‘

[Baldauf]

4 branches:1,2 = ..., inertial-gravity-waves3,4 0 false geostrophic mode with 4-point reconstruction (Nickovic et al., 2002)

Thuburn (2008), Thuburn, Ringler, Skamarock, Klemp (2009):8-point reconstruction of Coriolis term correct geostrophic mode 3,4 =0

Hexagon gridanalyticsolution

inertial-gravity wave

geostrophic mode

f

ka

[Baldauf]

ICON-project

• flux-form Semi-Lagrangian scheme (Miura, 2007) [Reinert], [Mittal]

from: D. Reinert (DWD)

Advection schemes

1. Rekonstruktion of the mass contained in this area

2. flux through the lower boundary

runs successfully in ICON

• Runge-Kutta based advection scheme [Skamarock]:

requirements: scheme for icosahedral hexagon grid, should match with the explicit Runge-Kutta time integration scheme

idea: higher order flux-formulation of Hundsdorfer et al. (1995) in 1D can be extended to unstructured grid by the reformulation

least-square fit to reconstruct the 2nd derivative

this produces much less phase/ampitude errors than 2nd order schemedramatically reduce phase errors in baroclinic wave test case in MPAS and ICON

Advection schemes

Thank you for attention

'Workshop on the Solution of Partial Differential Equations on the Sphere'24.-27. April 2010 in Potsdam, Germanyorganized by Alfred-Wegener-Institute (Bremerhaven, Potsdam ), Konrad-Zuse-Institute, Berlin

Purposes: A fundamental issue in climate and weather modeling is the dependence of errors in general circulation models on their basic dynamical cores and their subgrid scale parameterizations. To advance the state-of-the-art for numerical weather prediction and climate simulation, this organization encourages research on numerical algorithms and solution methods for partial differential equations (PDEs) posed in a spherical geometry. We also foster the development of tests and diagnostics for atmospheric and ocean model dynamical cores. The development of new methods in spherical geometry and how these eliminate shortcomings of current methods include applications to the shallow water equations and to complete hydrostatic and non-hydrostatic baroclinic models.

Topics of interests: 1. Advection schemes2. Discretization methods and adaptive grids3. Comparison study of methods4. Computational performance5. Discussion and proposal of test cases6. Global hydrostatic and non-hydrostatic models

Homepage of PDE's 2010

Topics of interests: 1. Advection schemes: monotone advection schemes, TVD methods, flux correction methods, and Eulerian and semi-Lagrangian methods applied to three dimensional baroclinic equations, two dimensional shallow water equations, and pure transport in spherical geometry and/or plane geometry.2. Discretization methods and adaptive grids: spectral transform methods, gridpoint methods using polyhedral grid systems such as icosahedral and cubed-sphere systems as well as the latitude longitude grids, spectral element methods, discontinuous Galerkin methods, and alternative discretizations on regular and unstructured adaptive grids.3. Comparison study of methods: the presentation of results from current or new methods, and the comparison of these methods with standard schemes.4. Computational performance: the performance of current methods of choice and new methods on parallel computers, including limitations associated with algorithm and computer design are appreciated; implementation details for distributed and shared memory architectures as well as discussion of attendant difficulties are of interest; characteristics for parallel computers that may affect computational algorithms, in particular, future design aspects could be considered as they might affect algorithm implementation.5. Discussion and proposal of test cases: the development and evaluation of new test methods, in particular, the potential ability of test cases to evaluate dynamical cores of atmospheric and ocean general circulation models; summaries of experiences with existing test cases, including what has been learned and what can be learned from such tests, and the introduction of additional test cases and experiences with them.6. Global hydrostatic and non-hydrostatic models: the development of global models based on the hydrostatic and non-hydrostatic equations; now that recent computer facilities enable us to perform very high-resolution simulation over the globe, we are in a period of transition of the choice of the equations system.

'Workshop on the Solution of Partial Differential Equations on the Sphere'24.-27. April 2010 in Potsdam, Germany

Semi-Implicit Semi-Lagrangian models

UK MetOffice: lat-lon-grid

Conformal-cubic atmosphere model (CCAM) [McGregor]cubed sphere; stretching Schmidt transformation

Russia: SL-AV model [Tolstykh]France: ALADIN model

multigrid solverfacilitated by the recursive data structure of the icosahedral grid

(used e.g. for anelastic eqns. in the CSU-model [Heikes])

Time integration methods

Split-Explicit modelsdistinguish slow (advection) and fast

(sound, gravity waves) dynamical processes different time steps

• VCAM [McGregor]• COSMO• NICAM [Tomita]

Time integration methods

Exact integration by Laplace-transformation of fast processes, no orographic resonance even with coupling with a Semi-Lagrangian transport scheme [Clancy] Lynch (1985)

Explicit (HE/VI) Models

•ICON-model [Zängl], [Gassmann]•MPAS [Klemp]