I. A pressure Poisson method for the incompressible Navier-Stokes equations : II. Long time behavior of the Klein-Gordon equations

Abstract

In this thesis, we address two problems involving partial differential equations. In the first problem, we reformulate the incompressible Navier-Stokes equations into an equivalent pressure Poisson system. The new system allows for the recovery of the pressure in terms of the fluid velocity, and consequently is ideal for efficient but also accurate numerical computations of the Navier-Stokes equations. The system may be discretized in theory to any order in space and time, while preserving the accuracy of solutions up to the domain boundary. We also devise a second order method to solve the recast system in curved geometries immersed within a regular grid. In the second problem, we examine the long time behavior of the Klein-Gordon equation with various nonlinearities. In the first case, we show that for a positive (repulsive) strong nonlinearity, the system thermalizes into a state which exhibits characteristics of linear waves. Through the introduction of a renormalized wave basis, we show that the waves exhibit a renormalized dispersion relation and a Planck-like energy spectrum. In the second case, we discuss the case of attractive nonlinearities. In comparison, here the waves develop oscillons as long lived, spatially localized oscillating fields. With an emphasis on their cosmological implications, we investigate oscillons in an expanding universe, and study their profiles and stability. The presence of a saturation nonlinearity results in flat-topped oscillons, which are relatively stable to long wavelength perturbations.