Numerical Analysis of Evolution Equations.
Final rept. 1 May 92-31 Oct 96,
STANFORD UNIV CA
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The overall objective of this work is to analyze and design effective computational algorithms for the integration of evolution equations over long time intervals. Many models of physical significance are characterised by the property of sensitive dependence on initial conditions small changes in the given data can make large changes in the detailed output of the model. Examples of such systems include weather or climate models in certain parameter regimes and turbulent flow problems. For such systems the effect of numerical approximation is not immediately clear. We may view numerical approximation as a small perturbation and the previous discussion indicates that this can nonetheless have a large effect on the detailed output from the model, over long time intervals. Thus it is important to know how to interpret data from such numerical simulations. Furthermore, in long-time integration, it is often crucial that the correct energy balance be used in the equation - be it dissipation or conservation. Thus it is important to design methods which replicate the energy balance in the equation under mild or no restrictions on the discretization parameters. These objective have been achieved and the following list of Awards, Invited Presentations, Graduated Students and Publications are all directly related to the support obtained through this grant.
- Numerical Mathematics