Improvement of the Semi-Lagrangian advection by ‘selective filling diffusion (SFD)'

WG2-meetingCOSMO-GM, Moscow, 06.09.2010

Michael Baldauf (FE13)

COSMO-Modell contains several methods for tracer advection:

• simple centered differences • Lin, Rood-scheme

In particular in combination with Runge-Kutta dynamical core:

• Bott-scheme (Finite Volume scheme)+ locally conserving (at least for C<1)- direction splitting of 1D-steps potential source of instabilities

• Semi-Lagrangian-scheme- not locally conserving+ relatively robust- sometimes 'stripe patterns' along coordinate lines occur- in singular points high precipitation values can occur

COSMO-EU'02.05.2010'0 UTC run24h-precipitation sum

SL with MF

COSMO-EU'02.05.2010'

SL with SFD

advection eq. (1-dim.)

rewritten as

Semi-Lagrangian-Advection

step 1: calculation of backward trajectory xjn-1

in principle any ODE-solver can be used (here: 2nd order)

Staniforth, Côté (1991) MWRBaldauf, Schulz (2004) COSMO-Newsl.

~

2nd step: Interpolation from neighbouring points

linear weighting polynomials:

cubic weighting polynomials:

x,y,z [0,1] = position in the grid cell (from backtrajectory calculation)qi,j,k = grid point value of q

Semi-Lagrangian Advection

i,j,k = -1,0 for tri-linear interpol. 8 grid pointsi,j,k = -2, ...,1 for tri-cubic interpol. 64 grid points

properties of Semi-Lagrangian advection

+ unconditionally stable (i.e. no CFL condition, but Lifshitz-condition) + fully multi-dimensional scheme (no directional splitting necessary quite robust)+ increased efficiency if used for many tracers (calculation of backtrajectory

only once)+ linear scheme, if used without clipping+ can be implemented also in unstructured grids+ no non-linear instability if used for velocity advection

- non-conserving scheme; but for higher order schemes conservation properties are not bad (without clipping):example: tri-cubic interpolation is exactly conserving in the case v=const (and cartesian grid)

- multi-cubic interpolation generates over-/undershoots not positive definitefor tracer advection: clipping of negative values necessary; this is a tremendous source of mass = strong violation of conservation

(multi-linear interpolation monotone, but highly diffusive)

1D-Advection with v=const (CFL=0.6)

exact solution

cubic interpol.without clipping

cubic interpol.with clipping

cubic interpol.with SFD

FE 13 – 22.04.23

Multiplicative Filling (Rood, 1987) SL - MF

clipped values are globally summed and distributed over the whole field

• easy • fast• but only global conservation

Problem of reproducibility: a sum of 'real' (=floating point) numbers is not associative:

(a + b) + c a + ( b + c )solution: a sum of integer numbers is associative

map the Real number space to the Integer number space( subroutine sum_DDI( field(:,:) ) in numeric_utilities_rk.f90 )

up to now:

but this is an unsatisfying solutionmoreover on massively parallel computers: a global operation is needed

PBPV – 03/2010

to get closer to local conservation:fill negative values from positive values from the environment

proposal: Semi-Lagrangian scheme with 'selective filling diffusion' (SFD)

1. tri-cubic interpolation2. artificial 3D-diffusion only in the vicinity of negative values

fills up negative values• diffusion itself can be formulated mass-conserving (FV)• diffusion is ‘well-tempered’:

only low requirements to the accuracy of the flux calculation, relativiely efficient

3. if grid points with negative values remain clipping

1D-Advektion mit v=const (CFL=0.6)

exact solution

cubic interpol.without clipping

cubic interpol.with clipping

cubic interpol.with SFD

Idealised advection tests (with prescribed v-field) in the COSMO-Model

Initialisierung '3D-Kegel-fkt.'

in the following plots:

difference against the analytic solution

initial distribution:3D-cone

SL - MF SL- SFD

SL - clip Bott

Test 1: advection with v=const in terrain following grid (CFL=0.107)

PBPV – 03/2010

SL with Clipping:5% mass increase!

Bott: exactly conserving

SL with 'SFD':0.2% mass increase

Test 1: advection with v=const in terrain following grid (CFL=0.107)

PBPV – 03/2010

SL with clipping:2.7% mass increase!

Bott: 0.1% mass increase

SL with 'SFD':0.15% mass increase

Test 2: advection with v=const in terrain following grid (CFL=1.5

Test 3: Solid body rotation test

= (-3.5, -3.5, 280) * const ( 1 turn around in 2 h)

initial field: 3D-cone

Test 3: Solid body rotation test = (-3.5, -3.5, 280) * const ( 1 turn around in 2 h)

SL - MF SL- SFD

SL - clip Bott

Test 3: Solid body rotation test = (-3.5, -3.5, 280) * const ( 1 turn around in 2 h)

SL - MF SL- SFD

SL - clip Bott

SL with clipping:8.5% mass increase!

Bott: exactly conserving

SL with 'SFD'0.7% mass increase

Conservation in the solid body rotation test

Test 4: 'LeVeque'-test (initial field: 3D-sphere)

crashed

SL - MF SL- SFD

SL - clip Bott

Synop-Verification: COSMO-EU (7km) 27.07.-27.08.2010 red: SL with SFDblue: SL with MF

Synop-Verification: COSMO-EU (7km) 27.07.-27.08.2010 red: SL with SFDblue: SL with MF

PBPV – 03/2010

Summary

‘selective filling diffusion (SFD)’ in the Semi-Lagrangian scheme• improves local conservation properties (if non-negativeness is needed)• often the 'best' scheme in idealised advection experiments• ‘multiplicative filling’ no longer needed (but could be applied afterwards)

• improves linear properties of the tracer-advection• synop-verification COSMO-EU (7km) (for 'August 2010'):

• small (but probably insignificant) improvements in RMSE• slightly higher biases

• in general 'stripe-patterns' and tendency to spots with high precipitation hasnot improved

outlook:• some tuning of the SFD necessary (?) (Thresholds)• Efficiency on vector computers (NEC SXx):

• 'diffusion in only a few points' ? 'diffusion everywhere with a lot K=0' ?• tri-cubic interpolation not optimised for the NEC-SX9

(vectorisation degree is 99.8%, but a lot of bank conflicts)

Initialisation '3D-sphere'