Efficient numerical algorithms for simulating evolution equations

In this dissertation, we concentrate on four distinct topics that focus on developing efficient numerical algorithms for evolving nonlinear PDEs. We are interested in studying the long-time behavior of these systems. The first three topics focus on gaining a better understanding, and improving existing algorithms for parabolic dissipative PDEs. The last topic involves non-dissipative equations and concentrates on implementing a general algorithm for simulating equations with multiple time scales.; In Chapter 2, we extend the idea of post-processing numerical schemes to nonlinear Galerkin and filtered Galerkin/nonlinear Galerkin numerical methods. Computations show the post-processed filtered Galerkin method is the most efficient method for the Fourier spectral case. We revisit the post-processing algorithm in Chapter 3, and give a different justification for this algorithm from a classical truncation analysis point of view. From this analysis we show that the post-processed Galerkin method is the leading order approximation method. In Chapter 4 we investigate some of the issues raised by J. G. Heywood and R. Rannacher [31] regarding the mechanism of improved accuracy for nonlinear Galerkin methods. In particular, we study a truncated nonlinear Galerkin method introduced in [31] for a one-dimensional linear diffusion transport equation. We show the nonlinear Galerkin methods do more than treat a boundary incompatibility. Finally, in Chapter 5, we study the Method of Averages (MOA) algorithm, introduced in [46] for numerically solving multiple time scale evolution problems. We investigate the effectiveness of the MOA for integrating a variety of geophysical systems.