Analysis of splitting methods for time-stepping physical parametrizations

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Analysis of splitting methods for time-stepping
physical parametrizations

• M. Dubal, N. Wood A. Staniforth
• Met Office, Exeter, U.K.

Resolved Dynamics Semi-implicit
semi-Lagrangian Large time-step Efficient numerics
NWP Model Must be robust accurate Coupling
problems Large time-step problems
Physics Complex multi-timescale Parametrization
s interactions Numerics poorly understood

• Couple dynamics and physics using splitting
  methods for simplicity and low cost
• Is there an optimal splitting strategy?
• Should we use parallel or sequential splittings
  of physics components?
• Should we use explicit or implicit numerical
  methods (or a mixture)?

Parallel Splitting Efficient numerics Modular
design No interactions of physics processes
Sequential Splitting Can have interactions of
physics processes Best sequential order to
include each process?
Consider a model problem
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A model problem
Parallel Splitting Method Find tendency from
each physics process alone Sum each tendency to
find total tendency Physics processes are unaware
of each other
Concurrent Method Standard finite-difference
technique All terms treated simultaneously Too
difficult and expensive to be useful Correct
steady-state obtained
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A model problem (continued)
Sequential Splitting Method Physics tendencies
added one after the other Total tendency updated
after each contribution Physics process sees
previously included processes In what order
should the physics be included? Scheme can be
symmetrized There is an
alternative symmetrized scheme
Sequential Splitting Method (2)
The sequential ordering can be
important Process ordering does not commute in
general
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Multiple time-scale problem An example
Aim solve differential equation numerically Use
time-steps of O(1) Explicit methods will be of
little use Solve the slow part accurately with
fast part present Coupling method should model
reduced system accurately Initialization of data
is an issue The fast mode should not be excited
Parallel Splitting Method Compute each tendency
alone Add to find total tendency
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Multiple time-scale problem (2)
Sequential Splitting Method

Initialization
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Accuracy
Second-order vs. First-order Want good accuracy
for very large time-steps
Summary Parallel sequential splitting
analysed Splitting errors produce time-step
dependent results Explicit parallel-split is
equivalent to concurrent Explicit parallel-split
is often unstable Non-explicit parallel-split
methods lead to error Sequential splitting
appears more flexible Splitting demonstrated for
multiple time-scale problem For stiff problems
initialization is an issue Damping of first-order
schemes can be useful Benefit of second-order
only realized if initialization is good
References

• Caya A., Laprise R. Zwack P. Mon. Wea. Rev.
  126, 1707-1713, (1998)
• Cullen M. Salmond D. Q.J.R. Meteorol. Soc.
  129, 1217-1236, (2002)
• Dubal M., Wood N. Staniforth A. Mon. Wea. Rev.
  (in press)
• Staniforth A., Wood N. Cote J. Q.J.R.
  Meteorol. Soc. 128, 2779-2799, (2002)
• Staniforth A., Wood N. Cote J. Mon. Wea. Rev.
  130, 3129-3135, (2002)

E-mail mark.dubal_at_metoffice.com