Abstract
This thesis is divided into two parts. In the first part we deal with numerical orthogonalisation and analyticity in hydrodynamic stability. We implemented a family of structure-preserving integrators and used them for accurate computations of the eigenvalues of the Orr-Sommerfeld equation. We address the problem of loss of analyticity induced by the use of geometric integrators from both an analytic and numerical perspective. We show that the loss of analyticity can be overcome when counting the number of eigenvalues in the interior of a bounded domain of the complex plane. The second part of the thesis deals with computation of defects in reaction-diffusion systems. Reaction-diffusion systems posed on the real line can exhibit solutions formed by wave trains at x = -infinity and X = +infinity which are connected by an intermediate interface region: such solutions are referred to as defects. Continuation of defects involves solving a boundary-value problem in an appropriate frame of reference. We discuss a method for the discretisation of such problems and implement it into ParaCont, a software for continuing large system of nonlinear equations. Finally we apply this method to continue defects in the Brusselator model and stationary localised solutions to the cubic-quintic Swift-Hohenberg equation.