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This thesis explains and tests a wavelet based implicit numerical method for the solving of partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time discretization into multiple, smaller, systems that can be solved sequentially. Included are tests of this method on linear and non-linear problems, with both its results and the time required to calculate them compared to basic models. It was found that the method requires less computational effort than the high resolution control results. Furthermore, the method showed convergence towards high resolution control results.

We study behavior for negative times t of the 2D periodic Navier-Stokes equations and Burgers' original model for turbulence. Both systems are proved to have rich sets of solutions that exist for all t - R and increase exponentially as t -> -(Infinity) However, our study shows that the behavior of these solutions as well as the geometrical structure of the sets of their initial data are very different. As a consequence, Burgers original model for turbulence becomes the first known dissipative system that despite possessing a rich set of backward-time exponentially growing solutions, does not display any similarities, as t -> -(Infinity), to the linear case.

This thesis explains and tests a wavelet based implicit numerical method for the solving of partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time discretization into multiple, smaller, systems that can be solved sequentially. Included are tests of this method on linear and non-linear problems, with both its results and the time required to calculate them compared to basic models. It was found that the method requires less computational effort than the high resolution control results. Furthermore, the method showed convergence towards high resolution control results.

This thesis explains and tests a wavelet based implicit numerical method for the solving of partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time discretization into multiple, smaller, systems that can be solved sequentially. Included are tests of this method on linear and non-linear problems, with both its results and the time required to calculate them compared to basic models. It was found that the method requires less computational effort than the high resolution control results. Furthermore, the method showed convergence towards high resolution control results.

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Road traffic networks offer a particularly challenging research subject to the control community. The traffic congestion around big cities is constantly increasing and is now becoming a major problem. However, the dynamics of a road network exhibit some complex behaviours such as nonlinearities, delays and saturation effects that prevent the use of some classical control algorithms. This thesis presents different models and control algorithms used for road traffic networks. The dynamics are represented using a "fluid-flow" approach. This leads to a system of quasi-linear hyperbolic partial differential equations which represents the behaviour of the drivers on each road. The boundary conditions are represented by a set of algebraic relations describing the behaviour of the drivers at the junctions. Two models with different complexities are introduced and their properties analysed. Different control algorithms are presented. One method is focused on the steady state case and intends to minimise a "sustainable cost" function. This function takes into account a time cost, the pollution and the accident risk. Two other methods which are able to deal with transient effects are also presented. The first one is a routing strategy expressing how to spread the traffic flow between two paths leading to the same destination. The second one is a ramp metering strategy using linear feedback.

This thesis presents a new preconditioner for elliptic PDE problems on unstructured meshes. Using ideas from second generation wavelets, a multi-resolution basis is constructed to effectively compress the inverse of the matrix, resolving the sparsity vs. quality problem of standard approximate inverses. This finally allows the approximate inverse approach to scale well, giving fast convergence for Krylov subspace accelerators on a wide variety of large unstructured problems. Implementation details are discussed, including ordering and construction of factored approximate inverses, discretization and basis construction in one and two dimensions, and possibilities for parallelism. The numerical experiments in one and two dimensions confirm the capabilities of the scheme. Along the way I highlight many new avenues for research, including the connections to multigrid and other multi-resolution schemes.

We investigate the local and global well-posedness of a variety of nonlinear Dirac type equations with null structure on R1+1. In particular, we prove global existence in L2 for a nonlinear Dirac equation known as the Thirring model. Local existence in Hs for s > 0, and global existence for s > 1/2 , has recently been proven by Selberg-Tesfahun where they used Xs,b spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara-Nakanishi-Tsugawa, we prove local existence in the scale invariant class L2 by using null coordinates. Moreover, again using null coordinates, we prove almost optimal local wellposedness for the Chern-Simons-Dirac equation which extends recent work of Huh. To prove global well-posedness for the Thirring model, we introduce a decomposition which shows the solution is linear (up to gauge transforms in U(1)), with an error term that can be controlled in L∞. This decomposition is also applied to prove global existence for the Chern-Simons-Dirac equation. This thesis also contains a study of bilinear estimates in Xs,b± (R2) spaces. These estimates are often used in the theory of nonlinear Dirac equations on R1+1. We prove estimates that are optimal up to endpoints by using dyadic decomposition together with some simplifications due to Tao. As an application, by using the I-method of Colliander-Keel-Staffilani-Takaoka-Tao, we extend the work of Tesfahun on global existence below the charge class for the Dirac-Klein- Gordon equation on R1+1. The final result contained in this thesis concerns the space-time Monopole equation. Recent work of Czubak showed that the space-time Monopole equation is locally well-posed in the Coulomb gauge for small initial data in Hs(R2) for s > 1/4 . Here we show that the Monopole equation has null structure in Lorenz gauge, and use this to prove local well-posedness for large initial data in Hs(R2) with s > 1/4.