1. Implementation of an Advection Scheme based on Piecewise Parabolic Method (PPM) in the MesoNH
The main goal of this study was to estimate if the RGSW model would be suitable for the description of the gap flow in the Wipp valley.

2. Introduction currently available advection schemes in MesoNH are:
centered 2nd order (CEN2ND) scheme for momentum advection
flux-corrected transport (FCT)
multidimensional positive definite advection transport algorithm (MPDATA)
leap-frog scheme used for time marching

3. Introduction
interested in implementing an accurate and more efficient advection scheme into the MesoNH
advection of a large number of chemical species
new, monotone, advection scheme would potentially operate on larger time step (separate from the model dynamics)

4. Introduction
semi-Lagrangian scheme tested for 2D (Stefan Wunderlich and J-P Pinty, 2004)
very accurate
allows for large time steps (works with Courant numbers greater than 1)
extension to 3D (vertical) non-trivial
parallelization and grid nesting…
open boundary conditions…
investigate another option, the PPM scheme

5. Introduction
as introduction for the PPM, centered 4th order advection scheme (CEN4TH) was prepared by J-P Pinty
now fully implemented (?)
works for all boundary conditions
parallelized
optional separate advection of momentum (U,V,W) and scalar fields with CEN4TH

6. PPM scheme introduced by Colella and Woodward in 1984
implemented and used in many atmospheric sciences and astrophysics applications (Carpenter 1990, Lin 1994, Lin 1996, … , also available in WRF, Skamarock 2005)
several modifications (e.g. extension to Courant numbers greater than 1) and improvements made

7. PPM algorithm:

8. piecewise parabolic polynomial
PPM algorithm:
piecewise parabolic polynomial

9. PPM algorithm:

10. PPM algorithm:

11. PPM algorithm:

12. PPM algorithm:

13. PPM scheme
to ensure that the scheme is monotonic, constraints are applied on parabolas’ parameters
positive definite: does not generate negative values from non-negative initial values
monotonic: does not amplify extrema in the initial values
monotonic scheme is also positive definite and consistent

14. PPM scheme
Lin 1994 and 1996 suggests 3 different monotonic and semi-monotonic constraints:
fully monotonic - PPM_01
“semi-monotonic” - PPM_02 - eliminates only undershoots
“positive definite” - PPM_03 - eliminates only negative undershoots
it is possible to use non-monotonized version (e.g. in WRF) - PPM_00

15. PPM scheme fully monotonic 1D PPM periodic BC Δx = 1, nx = 100
shape advected through the domain 5 times
PPM_01

16. PPM scheme
semi -monotonic 1D PPM
PPM_02

17. PPM scheme
positive definite D PPM
PPM_03

18. Implementing the PPM in MesoNH (2D)
PPM algorithm requires forward in time integration, not leap-frog
several ways to adapt the leap-frog scheme to work with the PPM advection:

19. Implementing the PPM in MesoNH (2D)

20. Implementing the PPM in MesoNH (2D)
operator splitting following Lin 1996: 3 1 2

21. MesoNH setup for the PPM scheme testing
2D idealized-flow tests with passive tracer transport in horizontal plane
Cartesian grid (100 x 100 x 1) with Δx = Δy = 1
prescribed stationary flow
periodic (CYCL) boundary conditions
numerical diffusion and Asselin time filter switched off
single-grid calculation on 1 CPU Linux PC

22. Testing the PPM – simple rotation, ω = const.
one full rotation in 1200 s
max Courant number = 0.37
average courant number = 0.2
advecting cone-shaped tracer field

23. Testing the PPM – simple rotation, ω = const.
FCT

24. Testing the PPM – simple rotation, ω = const.
MPDATA

25. Testing the PPM – simple rotation, ω = const.

26. Testing the PPM – simple rotation, ω = const.

27. Testing the PPM – simple rotation, ω = const.

28. Simple rotation – diagnostics

29. Simple rotation – diagnostics

30. Simple rotation – diagnostics

31. Simple rotation – diagnostics
error analysis following Takacs 1985

32. Simple rotation – diagnostics

33. Stability of the advection schemes
PPM schemes stable up to Courant numbers max(Cx,Cy) = 1
this is verified for MesoNH with advection only
FCT and MPDATA schemes become unstable at much smaller Courant numbers (less than 0.35 for MPDATA)
CEN4TH also unstable for C > 0.4, but theoretically should be stable for Courant numbers up to 0.72
perhaps because of different advection operator splitting?

34. Work in progress
incorporate the PPM scheme for scalar advection into the full 3D model
some problems with time marching ?
implement OPEN boundary conditions into the PPM scheme
continue working on semi-Lagrangian scheme (extension to 3D)

35. Summary new centered 4th order scheme CEN4TH implemented
should be used for momentum advection in combination with e.g. FCT2ND for scalars
several versions of monotone and semi-monotone PPM schemes in implementation
better accuracy and stability properties than existing schemes
still need to be fully implemented into the MesoNH

36. Questions?

38. PPM algorithm:

39. PPM scheme fully monotonic with steepening 1D PPM
fairly complicated and numerically expensive procedure
PPM_1S

40. Testing the PPM – cyclogenesis, ω(r)
max Courant number = 0.32
average Courant number = 0.1

41. Testing the PPM – cyclogenesis, ω(r)
FCT

42. Testing the PPM – cyclogenesis, ω(r)
MPDATA

43. Testing the PPM – cyclogenesis, ω(r)

44. Testing the PPM – cyclogenesis, ω(r)

45. Testing the PPM – cyclogenesis, ω(r)
with steepening

46. Stability of the advection schemes
the PPM schemes should be stable for Courant numbers up to one, Cr = 1
CEN4TH with leap-frog time marching should be stable up to Cr = 0.72
simple test: advection along diagonal with uniform flow speed (u = v = 0.25), varying Δt

47. Stability of the advection schemes
advection along the diagonal, from bottom left to top right corner
u = v = 0.25 m/s
for Δt = 1, Cx = Cy = 0.25
PPM schemes should work for up to Δt = 5

48. Stability of the advection schemes
FCT
Cx,y=0.25
C = 0.35
PPM_01
Cx,y = 1
C = 1.41
MPDATA
Cx,y=0.25
C = 0.35

49. Future work implement open boundary conditions for the PPM schemes
parallelize the code
implement new time-marching scheme, RK3 (better accuracy, larger Cr, full use of the PPM schemes) ?
further investigate the stability issues of CEN4TH, FCT and MPDATA schemes ?