Implementation of an Advection Scheme based on Piecewise Parabolic

Method (PPM) in the MesoNH

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 algorithm:

PPM algorithm:

piecewise parabolic polynomial

PPM algorithm:

PPM algorithm:

PPM algorithm:

PPM algorithm:

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

PPM scheme

• semi -monotonic 1D PPM

PPM_02

PPM scheme

• positive definite 1D PPM

PPM_03

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:

Implementing the PPM in MesoNH (2D)

Implementing the PPM in MesoNH (2D)

• operator splitting following Lin 1996:

1 2

3

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

Testing the PPM – simple rotation, ω = const.

PPM_01 FCT

Testing the PPM – simple rotation, ω = const.

PPM_01 MPDATA

Testing the PPM – simple rotation, ω = const.

PPM_01 PPM_02

Testing the PPM – simple rotation, ω = const.

PPM_01 PPM_03

Testing the PPM – simple rotation, ω = const.

PPM_01 PPM_00

Simple rotation – diagnostics

Simple rotation – diagnostics

Simple rotation – diagnostics

Simple rotation – diagnostics

• error analysis following Takacs 1985

Simple rotation – diagnostics

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

Questions?

PPM algorithm:

PPM scheme

• fully monotonic with steepening 1D PPM

• fairly complicated and numerically expensive procedure

PPM_1S

Testing the PPM – cyclogenesis, ω(r)

max Courant number = 0.32

• average Courant number = 0.1

Testing the PPM – cyclogenesis, ω(r)

PPM_01 FCT

Testing the PPM – cyclogenesis, ω(r)

PPM_01 MPDATA

Testing the PPM – cyclogenesis, ω(r)

PPM_01 PPM_02

Testing the PPM – cyclogenesis, ω(r)

PPM_01 PPM_03

Testing the PPM – cyclogenesis, ω(r)

PPM_01 PPM_01 with steepening

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

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 ?