CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Splitting Methods for Problems with Different Time Scales

SPEAKER: Jerry Browning, Colorado State University  and NOAA Forecast 
         Systems Laboratory (joint work with Heinz Kreiss, UCLA)

DATE:    Wednesday, October 18, 1995

PLACE:   Math Conference Room - Suite 540
         UCD Building, 1250 14th St., Denver

TIME:    2:30 pm (Refreshments served at 2:15 pm)

ABSTRACT

The time step for the leapfrog scheme for a symmetric hyperbolic system 
with multiple time scales is limited by the Courant-Friedrichs-Lewy condition
based on the fastest speed present. However, in many physical cases, most of 
the energy is in the slowest wave, and for this wave the use of the above time
step implies that the time truncation error is much smaller than the spatial 
truncation error. A number of methods have been proposed to overcome this 
imbalance, e.g. the semi-implicit method and the additive splitting technique
originally proposed by Marchuk with variations attributable to Strang, Klemp 
and Wilhelmson. An analysis of the Marchuk splitting method for multiple time
scale systems shows that if a time step based on the slow speed is used, the
accuracy of the method cannot be proved and in practice the method is quite 
inaccurate. If a time step is chosen that is between the two extremes, then the
Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization
mechanism is added. The additional computational burden required to maintain 
the accuracy and the stability of the split-explicit method leads to the 
conclusion that it is no more efficient than the leapfrog method trivially 
modified to handle computationally expensive smooth forcing terms.

Using the mathematical analysis developed in a previous manuscript,
it is shown that splitting schemes are not appropriate for badly skewed 
hyperbolic systems. In a number of atmospheric models, the semi-implicit 
method is used to treat the badly skewed vertical sound wave terms. This
leads to the excitation of the high-frequency waves in a nonphysical manner.
It is also shown that this is equivalent to solving the primitive equations,
i.e. a model using this method for the large-scale case will be ill-posed at 
the lateral boundaries. The multiscale system for meteorology was introduced 
by Browning and Kreiss to overcome exactly these problems.