Multiscale Numerical Strategies for Models with Quadratic Nonlinearity

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The main goal of the proposed research is to develop a novel mathematical approach to allow faster numerical integration of certain types of Partial Differential Equations. The types of Partial Differential Equations considered in this proposal are very common in many areas of physics and, in particular, play a crucial role in numerical weather and climate studies. The main idea behind the proposed approach is that in many applications the main quantities of interest are large-scale averaged quantities (e.g., mean temperature changes over the next five to ten years, mean wind velocities during the summer, mean sea-surface temperature). Therefore, in such applications it is not necessary to resolve all small-scale physics (e.g., local wind speed at any particular location) accurately. On the other hand, the time-step of numerical integration is often limited (for technical reasons) by these small-scale processes. The proposed research seeks systematic modification of the underlying partial differential equations to reduce the overall influence of small-scale processes and, thus, to allow for a bigger time-step in numerical simulations.