flux form semi-lagrangian transport in icon: construction and results of idealised test cases

Deutscher Wetterdienst

Flux form semi-Lagrangian transport in ICON:construction and results of idealised test cases

Daniel Reinert

Deutscher Wetterdienst

Workshop on the Solution of Partial Differential Equations on the Sphere

Potsdam, 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


I. Flux form semi-Lagrangian scheme (FFSL) for horizontal transport


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


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.


control volume

1. Let this be our Eulerian control volume (area Ai), with area averages stored at the mass point

Physical/graphical interpretation ?


trajectories

control volume


2. Assume that we know all the trajectories terminating at the CV edges at n+1


control volume

3. Now we can construct the Lagrangian CV (known as „departure cell“)


In a ‚real‘ semi-Lagrangian scheme we would integrate over the departure cell


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


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.

26.08.2010

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:


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.

26.08.2010

Approximations according to Miura (2007)

unknown

mass point of control volume i


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


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


IV. Results: Linear vs. quadratic reconstruction


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°)


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


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


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


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


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.


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


Thank you for your attention !!


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.

flux form semi-lagrangian transport in icon: construction and results of idealised test cases

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