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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. - PowerPoint PPT Presentation
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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…