Since Patankar first developed the SIMPLE (Semi Implicit Method for Pressure Linked Equations) algorithm, the incompressible Navier-Stokes equations have been solved using a variety of pressure-based methods. Over the last twenty years these methods have been refined and developed however the majority of this work has been based around the use of structured grids to mesh the fluid domain of interest. Unstructured grids offer considerable advantages over structured meshes when a fluid in a complex domain is being modelled. By using triangles in two dimensions and tetrahedrons in three dimensions it is possible to mesh many problems for which it would be impossible to use structured grids. In addition to this, unstructured grids allow meshes to be generated with relatively little effort, in comparison to structured grids, and therefore shorten the time taken to model a particular problem. Also, through the use adaptive refinement, the mesh generation process can be coupled to the solution algorithm to allow the mesh to be refined in areas where complex flow patterns exist. Whilst the advantages to unstructured meshes are obvious they have inherent difficulties associated with them. The computational overheads of using an unstructured grid are increased and the discretisation process becomes more complex. Also, it is inevitable that some of the discretisation methods used as standard on structured grids, do not perform as accurately when used on an unstructured mesh. Therefore, the use of unstructured meshes in computational fluid dynamics (CFD) is still an area of active research. This thesis aims to investigate the use of unstructured meshes to solve the incompressible Navier-Stokes equations using the SIMPLE algorithm. A discretisation strategy drawing on the work of others is developed, that attempts to maintain the accuracy of the solution despite the discretisation problems that unstructured grids present. Particular attention is paid to the convective term in the momentum equations, which is often the cause of inaccuracy in pressure-based solvers. High order convective models, first developed for structured meshes, are adapted for use within an unstructured discretisation to ensure stable and bounded solutions are calculated. To reduce computational costs, the discretisation is based on a pointer system that aims to minimise the amount of connectivity data stored for a particular grid. In addition an efficient multigrid algorithm accelerates the solution of the equations to achieve more realistic calculation times. As an initial test of the solver's accuracy and efficiency, calculated results are compared with standard laminar flow problems in both two and three dimensions. However, for any solution strategy to be of practical use it must be able to model turbulent flow. To that end the algorithm is extended to find solutions to the incompressible Reynolds averaged Navier-Stokes equations, using the k-? turbulence model to close the equations. Again, two and three-dimensional problems are used to test the solver's accuracy and efficiency at calculating turbulent flow. Finally the findings of the research work are summarised and conclusions drawn.

This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.

Having infinitely many generalised symmetries is one of the definition of integrability for non-linear differential-difference equations. Therefore, it is important to develop tools by which we can produce these quantities and guarantee the integrability. Two different methods of producing generalised symmetries are studied throughout this thesis, namely recursion operators and master symmetries. These are objects that enable one to obtain the hierarchy of symmetries by recursive action on a known symmetry of a given equation. Our first result contains new Hamiltonian, symplectic and recursion operators for several (1 + 1 )-dimensional differential-difference equations both scalar and multicomponent. In fact in chapter 5 we give the factorization of the new recursion operators into composition of compatible Hamiltonian and symplectic operators. For the list of integrable equations we shall also provide the inverse of recursion operators if it exists. As the second result, we have obtained the master symmetry of differentialdifference KP equation. Since for (2+1 )-dimensional differential-difference equations recursion operators take more complicated form, " master symmetries are alternative effective tools to produce infinitely many symmetries. The notion of master symmetry is thoroughly discussed in chapter 6 and as a result of this chapter we obtain the master symmetry for the differential-difference KP (DDKP) equation. Furthermore, we also produce time dependent symmetries through sl(2, C)-representation of the DDKP equation.

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