1. Semi-implicit predictor-corrector methods for atmospheric models
Colm Clancy
Janusz A. Pudykiewicz
Atmospheric Numerical Weather Prediction Research,
Environment Canada
PDEs on the Sphere, 26th of September 2012

2. Motivation
Development of a finite-volume atmospheric model on an icosahedral grid (Pudykiewicz 2006, 2011)
Investigation of stable time integration schemes, beyond the traditional semi-implicit leapfrog
Some recent work: Williams (2011), Durran & Blossey (2012), Kar (2012)

3. General ODE system

4. General ODE system
‘Traditional’ semi-implicit, (SILF):

5. Semi-implicit predictor-corrector approach
Predictor stage, for :
Corrector stage, for :

6. Implicit linear terms:
Trapezoidal
AM2*

7. Explicit nonlinear terms:

8. Many possible combinations…
Examples:

9. Linear stability analysis

10. Reference semi-implicit

12. Shallow water tests Shallow water model of Pudykiewicz (2011)
Iterative GCR(4) solver for Helmholtz equations (Smolarkiewicz and Margolin, 2000)
No explicit diffusion
Filter of Williams (2011) for the semi-implicit leapfrog
Spatial resolution: grid 6 (40,962 nodes, ~112km). Reference: grid 7 (163,842 nodes, ~56km) with RK4 at 90s time-step

13. Sample results: Flow over isolated mountain

14. Williamson et al. (1992) – Mountain case

15. Williamson et al. (1992) – Mountain case

16. Williamson et al. (1992) – Mountain case

17. Sample results: Rossby-Haurwitz wave

18. Williamson et al. (1992) – RH wave case

19. Williamson et al. (1992) – RH wave case

20. Williamson et al. (1992) – RH wave case

21. Efficiency
Predictor-corrector schemes: two elliptic solver calls per time-step
Consider total number of iterations per step

22. Efficiency

23. Conclusions and further work
Semi-implicit predictor-corrector schemes offer an accurate alternative to the traditional leapfrog:
Stable
No time filter necessary
Efficiency not affected
Future tests with a three-dimensional baroclinic model
Comparison with other time integration methods

24. References Clancy & Pudykiewicz (2012); to appear in J. Comp. Phys.
Durran & Blossey (2012); Mon. Weather. Rev. 140,
Kar (2012); Mon. Weather. Rev. 134,
Pudykiewicz (2006); J. Comp. Phys. 213,
Pudykiewicz (2011); J. Comp. Phys. 230,
Smolarkiewicz & Margolin (2000). Proc. ECWMF Workshop, 5-7 June 2000,
Williams (2011); Mon. Weather. Rev. 139,
Williamson et al. (1992); J. Comp. Phys. 102,

25. Many possible combinations…