Semi-implicit predictor-corrector methods for atmospheric models

Colm ClancyJanusz A. Pudykiewicz

Atmospheric Numerical Weather Prediction Research,Environment Canada

PDEs on the Sphere, 26th of September 2012

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)

General ODE system

General ODE system

‘Traditional’ semi-implicit, (SILF):

Semi-implicit predictor-corrector approach

Predictor stage, for :

Corrector stage, for :

Implicit linear terms:

Trapezoidal

AM2*

Explicit nonlinear terms:

Many possible combinations…

Examples:

Linear stability analysis

Reference semi-implicit

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

Sample results:

Flow over isolated mountain

Williamson et al. (1992) – Mountain case

Williamson et al. (1992) – Mountain case

Williamson et al. (1992) – Mountain case

Sample results:

Rossby-Haurwitz wave

Williamson et al. (1992) – RH wave case

Williamson et al. (1992) – RH wave case

Williamson et al. (1992) – RH wave case

• Predictor-corrector schemes: two elliptic solver calls per time-step

• Consider total number of iterations per step

Efficiency

Efficiency

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

References

• Clancy & Pudykiewicz (2012); to appear in J. Comp. Phys.

• Durran & Blossey (2012); Mon. Weather. Rev. 140, 1307-1325

• Kar (2012); Mon. Weather. Rev. 134, 2916-2926

• Pudykiewicz (2006); J. Comp. Phys. 213, 358-390

• Pudykiewicz (2011); J. Comp. Phys. 230, 1956-1991

• Smolarkiewicz & Margolin (2000). Proc. ECWMF Workshop, 5-7 June 2000, 137-159

• Williams (2011); Mon. Weather. Rev. 139, 1996-2007

• Williamson et al. (1992); J. Comp. Phys. 102, 211-224

Many possible combinations…