The fluids that make up our atmosphere and oceans exhibit extremely complex dynamics that are characterized by vast differences in scale, sensitive dependence on initial conditions, and scale-invariant solutions. Simulation of these systems is a challenging task, one that is continually being carried out---to predict the weather on the short scale, and the climate on the long scale.
An example is the Dutch Challenge Project at KNMI, in which the climate has been simulated over an interval of 140 years to investigate if increased global worming will have an adverse effect on extreme weather conditions. A climate model typically consists of the coupled atmospheric and oceanic fluids, supplemented with models for land and ice, solar radiation, water vapor and clouds, atmospheric chemical reactions and aerosols.
To account for uncertainty in the initial conditions at the start date of January 1940, the Dutch Challenge Project simulates the climate from 62 slightly perturbed initial states, i.e. an ensemble simulation. These simulations form the basis for a statistical analysis. For example, the figure here shows the average temperature per year at grid point "de Bilt", for each of 31 runs (in black). Superimposed on this is the ensemble average (in red) and the actual measured annual mean temperature in de Bilt since 1940 (in blue).
Since the work of E.N. Lorenz in the 1960s, it has been understood that the time-evolution of the atmospheric equations is unstable with respect to small changes in the initial state. For example, the figure here illustrates how the daily mean temperature in de Bilt varies for 31 simulations over the first month of simulation. This sensitivity is referred to as chaos and has received a lot of attention in the popular scientific press.
Formation of a discrete model (discretization)---a step required to simulate the atmospheric equations on a computer---necessarily introduces small errors in the solution. These errors will grow in the same way as the small perturbations to the initial state just mentioned. Eventually the errors will dominate and the simulated solution will look nothing like the solution that would have been obtained in the absence of discretization errors.
Obviously, the scientists involved in the Dutch Challenge Project do not expect their global climate simulation to accurately predict the weather on a Tuesday in 2080. Rather, they wish to use the simulation, and in particular the ensemble of simulations, to search for plausible and likely behavior given the best estimates to climatic input over the coming century.
A question of paramount importance is what one can say about the validity of a simulation over such a time scale in the presence of chaos.
The role of the discrete model is crucial. Since the errors grow out of control, one cannot hope to obtain an accurate solution in the strict numerical analytical sense. Instead one tries to ensure that the solution generated by the discrete model agrees in a statistical sense with the actual climate dynamics. The discrete model should attempt to provide an accurate representation of the phase space of the continuous problem. It should be locally accurate so that over relatively short intervals the dynamics are typical, and it should have excellent global properties to constrain the simulation within a physically relevant subset of the phase space.
All stable, convergent discretizations satisfy the local properties just mentioned. The identification and development of discrete models with good global properties is the subject of this research. Example of such properties are energy conservation, vorticity conservation, preservation of adiabatic invariants such as geostrophic and hydrostatic balances, symplecticity, ...