Formulation and numerical computation of the low frequency mean flow of fluids

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Highly oscillatory systems of partial differential equations (PDEs) dominate many scientific applications. We are now on the doorstep of a computing revolution with new, heterogeneous computer architectures which could enable significantly higher spatial resolution. One of the main issues standing in the way of taking advantage of these new computers is that as the spatial resolution increases, the time step must decrease. As an example, for an atmosphere model with a grid resolution of 300 kilometers, a 20 minute time step is generally used. Increasing the spatial resolution to 1 kilometer requires a reduction of the time step to 4 seconds, making high resolution simulations impractical for scientific exploration. However, it is known in theoretical fluid dynamics that low frequencies are created through nonlinear coupling of resonant and “near-resonant” waves. These frequencies dominate the dynamics of weather and climate applications. New formulations of the oscillatory PDEs that expose the described resonance behaviour can be used to derive equations for the mean flow which describes the nonlinear resonant and “near-resonant” part of the problem, and this, in turn, leads to constructing numerical methods able to take larger time steps than previously possible. In this project, I propose building on results from theoretical fluid dynamics and numerical analysis to 1) develop new formulations of the mean flow which take both the resonant and the “near-resonant” frequencies into account and 2) develop and analyse new numerical schemes, that do not neglect the resonant and also the “near-resonant” frequencies, with the goal of taking large time steps beyond current limitations. These schemes will be investigated by means of numerical analysis and implemented in idealized domains, mainly within the application of weather and climate simulation. The numerical schemes that will be explored include parallel-in-time schemes, because such schemes can contribute to substantially speeding up computations on modern computer architectures. The developed fluid dynamics theory and numerical algorithms have the potential to significantly decrease the time-to-solution for weather and climate models on new computer architectures.