deutscher wetterdienst flux form semi-lagrangian transport in icon: construction and results of...
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Daniel Reinert I.Flux form semi-Lagrangian scheme (FFSL) for horizontal transportTRANSCRIPT
Deutscher Wetterdienst
Flux form semi-Lagrangian transport in ICON:construction and results of idealised test cases
Daniel ReinertDeutscher Wetterdienst
Workshop on the Solution of Partial Differential Equations on the SpherePotsdam, 26.08.2010
Daniel Reinert26.08.2010
Outline
I. Flux form semi-Lagrangian (FFSL) scheme for horizontal transport A short recap of the basic ideas
II. FFSL-Implementation approximations according to Miura (2007) Aim: higher order extension
III. Results: Linear vs. quadratic reconstruction 2D solid body rotation 2D deformational flow
IV. Summary and outlook
Daniel Reinert26.08.2010
I. Flux form semi-Lagrangian scheme (FFSL) for horizontal transport
Daniel Reinert26.08.2010
FFSL: A short recap of the basic ideas
Scheme is based on Finite-Volume (cell integrated) version of the 2D continuity equation. Assumption for derivation: 2D cartesian coordinate system Starting point: 2D continuity equation in flux form
Problem: Given at time t0 we seek for a new set of at time t1= t0+Δt as an approximate solution after a short time of transport.
Control volume (CV): triangular cells Discrete value at mass point is defined to be the
average over the control volume
Daniel Reinert26.08.2010
In general, the solution can be derived by integrating the continuity equation over the Eulerian control volume Ai and the time interval [t0,t1].
FFSL: A short recap of the basic ideas
No approximation as long as we know the subgrid distribution ρq and the velocity field (or aie) analytically.
applying Gauss-theorem on the rhs, assuming a triangular CV, …… we can derive the following FV version of the continuity equation.
Daniel Reinert26.08.2010
control volume
1. Let this be our Eulerian control volume (area Ai), with area averages stored at the mass point
Physical/graphical interpretation ?
Daniel Reinert26.08.2010
trajectories
control volume
Physical/graphical interpretation ?
2. Assume that we know all the trajectories terminating at the CV edges at n+1
Daniel Reinert26.08.2010
control volume
3. Now we can construct the Lagrangian CV (known as „departure cell“)
Physical/graphical interpretation ?
In a ‚real‘ semi-Lagrangian scheme we would integrate over the departure cell
Daniel Reinert26.08.2010
We apply the Eulerian viewpoint and do the integration just the other way around:
Compute tracer mass that crosses each CV edge during Δt.
Material present in the area aie which is swept across corresponding CV edge
Physical/graphical interpretation ?
Daniel Reinert
1. Determine the departure region aie for the eth edge
2. Determine approximation to unknown tracer subgrid distribution for each Eulerian control volume
3. Integrate the subgrid distribution over the (yellow) area aie.
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Basic algorithm
example for edge 1
Note: For tracer-mass consistency reasons we do not integrate/reconstruct ρ(x,y)q(x,y) but only q(x,y). Mass flux is provided by the dynamical core.
The numerical algorithm to solve for consists of three major steps:
Daniel Reinert26.08.2010
II. FFSL-implementation
Daniel Reinert
1. Departure region aie: aproximated by rhomboidally shaped area. Assumption: v=const on a given edge
2. Reconstruction of qn(x,y): SGS tracer distribution approximated by 2D first order
(linear) polynomial.
conservative weighted least squares reconstruction
3. Integration: Gauss-Legendre quadrature No additional splitting of the departure region. Polynomial
of upwind cell is applied for the entire departure region.
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Approximations according to Miura (2007)
unknown
mass point of control volume i
Daniel Reinert26.08.2010
Default: first order (linear) polynomial (3 unknowns)
Stencil for least squares reconstruction
Possible improvement - Higher order reconstruction
Test: second order (quadratic) polynomial (6 unknowns)
3-point stencil 9-point stencil
improvements to the departure regions appear to be too costly for operational NWP
we investigated possible advantages of a higher order reconstruction.
CVCV
Daniel Reinert26.08.2010
3-point stencilGnomonic projection
Plane of projectionEquator
How to deal with spherical geometry ?
All computations are performed in local 2D cartesian coordinate systems We define tangent planes at each edge midpoint and cell center Neighboring points are projected onto these planes using a gnomonic projection.
Great-circle arcs project as straight lines
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IV. Results: Linear vs. quadratic reconstruction
Daniel Reinert26.08.2010
Uniform, non-deformational and constant in time flow on the sphere.
Initial scalar field is a cosine bell centered at the equator
After 12 days of model integration, cosine bell reaches its initial position
Analytic solution at every time step = initial condition
Solid body rotation test case
Error norms (l1, l2, l∞) are calculated after one complete revolution for different resolutions
Example of flow in northeastern direction (=45°)
Daniel Reinert26.08.2010
Setup R2B4 (≈ 140km) CFL≈0.5 =45° flux limiter conservative
reconstruction
L1 = 0.392E-01L2 = 0.329E-01
L1 = 0.887E-01L2 = 0.715E-01
Shape preservation
Errors are more symmetrically distributed for the quadratic reconstruction.
quadratic
linear
Daniel Reinert26.08.2010
Setup R2B4 (≈ 140km) CFL≈0.5 =45° flux limiter non-conservative
reconstruction
quadratic
linear
L1 = 1.293E-01L2 = 1.133E-01
L1 = 0.854E-01L2 = 0.699E-01
Non-conservative reconstruction
Conservative reconstruction: important when using a quadratic polynomial
Daniel Reinert26.08.2010
Convergence rates (solid body)
Quadratic reconstruction shows improved convergence rates and reduced absolute errors.
quadratic, conservative linear, non-conservative
Setup: CFL≈0.25 =45° (i.e. advection in northeastern direction) flux limiter
Daniel Reinert26.08.2010
Deformational flow test case
based on Nair, D. and P. H. Lauritzen (2010): A class of Deformational Flow Test-Cases for the Advection Problems on the Sphere, JCP
Time-varying, analytical flow field
Tracer undergoes severe deformation during the simulation
Flow reverses its course at half time and the tracer field returns to the initial position and shape
Test suite consists of 4 cases of initial conditions, three for non-divergent and one for divergent flows.
t=0 T t=0.5 T t=T
Example: Tracer field for case 1
Daniel Reinert26.08.2010
Convergence rates (deformational flow)
C≈0.50 flux limiter
quadratic, conservative linear, non-conservative
Superiority of quadratic reconstruction (absolute error, convergence rates) less pronounced as compared to solid body advection. But still apparent for l∞.
Possible reason: departure region approximation (rhomboidal) does not account for flow deformation.
Daniel Reinert26.08.2010
V. Summary and outlook
Implemented a 2D FFSL transport scheme in ICON (on triangular grid)
Based on approximations originally proposed by Miura (2007) for hexagonal grids
Pursued higher order extension of the 2nd order ‚Miura‘ scheme by using a higher order (i.e. quadratic) polynomial reconstruction.
Quadratic reconstruction led to improved shape preservation and reduced maximum errors improved convergence rates (in particular L∞) reduced dependency of the error on CFL-number
For more challenging deformational flows, the superiority was less marked
conservative reconstruction is essential when using higher order polynomials
Outlook: Which reconstruction (polynomial order) is the most efficient one?
Possible gains from cheap improvements of the departure regions?
Implementation on hexagonal grid comparison
Daniel Reinert26.08.2010
Thank you for your attention !!
Daniel Reinert26.08.2010
Model error as a function of CFL (solid body)
Fixed horizontal resolution: R2B5 (≈ 70 km) variable timestep variable CFL number flux limiter
Flow orientation angle: α=0° Flow orientation angle: α=45°
Linear rec.: increasing error with increasing timestep and strong dependency on flow orientation angle.
Quadratic rec.: errors less dependent on timestep and flow angle.