Semiimplicit predictorcorrector methods for atmospheric models Colm Clancy

Semi-implicit predictor-corrector methods for atmospheric models Colm Clancy Janusz A. Pudykiewicz Atmospheric Numerical Weather Prediction Research, Environment Canada PDEs on the Sphere, 26 th 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 AM 2*

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, ~112 km). Reference: grid 7 (163, 842 nodes, ~56 km) with RK 4 at 90 s 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

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

Efficiency

Conclusions and further work • Semi-implicit predictor-corrector schemes offer an accurate alternative to the traditional leapfrog: Ø Stable Ø No time filter necessary Ø 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…