Analysis of splitting methods for time-stepping physical parametrizations

1. Analysis of splitting methods for time-stepping physical parametrizations 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 Parametrizations & 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? M. Dubal, N. Wood & A. Staniforth Met Office, Exeter, U.K. Consider a model problem

2. x for any value of 1 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

3. 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

4. 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

5. Multiple time-scale problem (2) Sequential Splitting Method Initialization

6. 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@metoffice.com