| In this dissertation we study temporal integration techniques that are suitable to be applied in conjunction with the so-called Nonlinear Galerkin method when considering a collocation spectral discretization for the spatial operator. The main goal of this work is the application of distinct time marching schemes for the small and large scales of a dissipative system, which are known to have different physical and numerical behaviors. The first part of this dissertation provides a decomposition of the unknowns into low and high modes in the physical space, rather than the spectral one, for two dimensions in space and in the case of Fourier and Chebyshev collocation methods. The second part is the presentation of a new class of explicit and unconditionally stable schemes for the temporal integration of the high modes component. |
