Post on 31-Dec-2015
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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
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)
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
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
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
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
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
PPM scheme
• fully monotonic 1D PPM
• periodic BC• Δx = 1, nx =
100• shape advected
through the domain 5 times
PPM_01
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:
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
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
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?
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)
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
PPM scheme
• fully monotonic with steepening 1D PPM
• fairly complicated and numerically expensive procedure
PPM_1S
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
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
Stability of the advection schemes
PPM_01Cx,y = 1C = 1.41
FCTCx,y=0.25C = 0.35
MPDATACx,y=0.25C = 0.35