Model reduction for multi-scale partial differential equations
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In this thesis, a new method is developed for model reduction in multi-scale systems of coupled reaction-diﬀusion equations. The new method uses scaling variables which are naturally suggested by the classical diﬀusion operator. The main diﬀerential operators have a spectral gap and an inﬁnite basis of natural modes associated to the point spectrum. The ﬁrst principal results consist of establishing the existence of nonlinear slow modes for reaction-diﬀusion systems and of rigorously studying the algebraic and exponential decay of general solutions toward them. The new method exploits separations of time scales between slow and fast species. It is illustrated on two prototypical examples, the Davis-Skodje model and the Michaelis-Menten-Henri (MMH) model with diﬀusion of both species. The solutions of the Davis-Skodje and MMH models are decomposed completely into slow parts, which we label as the nonlinear slow mode, and fast parts which exhibit short term algebraic decay and long term exponential decay. The second principal result consists of introducing a new class of multi-scale reaction-diﬀusion equations that possess closed-form, lowdimensional, invariant manifolds. This new class of PDEs is of interest in its own right, since there are currently very few examples of PDEs known to have manifolds expressed in closed form. Also, it provides a useful set of benchmark problems for testing and comparing numerical methods for model reduction in nonlinear PDEs. The third principal result consists of a mathematical analysis of the Approximate Slow Invariant Manifold (ASIM) method developed by Powers and Paolucci. The accuracy of the ASIM method is proven for systems of reaction-diﬀusion equations with slow and fast reaction kinetics.