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The solution of convection-diffusion problems is a challenging task for numerical methods because of the nature of the governing equation, which includes a non-dissipative component and a dissipative component. Once the convection-diffusion equation is discretised, it is usual to observe oscillations in the computed solution regardless of whether these might be expected in the original physical situation. Mostly these oscillations are the result of numerical instability. This thesis centres on this fundamental difficulty: the numerical stability of finite difference discretisation of a convection-diffusion equation. The existence of an exact evolution operator for the constant coefficient convection diffusion problem is the framework we use to derive new finite difference schemes in one and two dimensions and also, when a high-order scheme is considered, to derive numerical boundary conditions. The influence of numerical boundary conditions on the stability of a general scheme is one of the main themes. The stability analysis is done mostly by using the von Neumann method and the matrix method. The Godunov-Ryabenkii theory is also applied to the one dimensional case. In two dimensions we deduce different forms of second-order (Lax-Wendroff) schemes and third-order (Quickest) schemes. We apply some of those schemes to a Navier-Stokes problem by running experiments to illustrate the practical stability region, showing how results from a simpler case presented in previous chapters carry over to the more complex case.

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Stock options are priced numerically using space- and time-adaptive finite difference methods. European options on one and several underlying assets are considered. These are priced with adaptive numerical algorithms including a second order method and a more accurate method. For American options we use the adaptive technique to price options on one stock with and without stochastic volatility. In all these methods emphasis is put on the control of errors to fulfill predefined tolerance levels. The adaptive second order method is compared to an alternative discretization technique using radial basis functions. This method is not adaptive but shows potential in option pricing for one and several underlying assets. A finite difference method and a Monte Carlo method are applied to a new financial contract called Turbo warrant. A comparison of these two methods shows that for the case considered the finite difference method is superior.

Thesis (Ph. D.)--University of Adelaide, Dept. of Applied Mathematics, 1994? / Includes bibliographical references (leaves 211-216).

Thesis (M. App. Sc.)--Carleton University, 2002. / Includes bibliographical references (p. 201-202). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1993. / Vita. Abstract. Includes bibliographical references (leaves 112-119). Also available via the Internet.

Thesis (M.S. in physics)--Washington State University, December 2008. / Title from PDF title page (viewed on July 10, 2009). "Department of Physics and Astronomy." Includes bibliographical references (p. 64-68).

Thesis (Ph.D..(Mathematics and Applied Mathematics)) -- University of Pretoria, 2009. / Summary and abstract in English. Includes bibliographical references.

Thesis (M.S.)--West Virginia University, 1999. / Title from document title page. Document formatted into pages; contains iv, 87 p. : ill. (some col.). Includes abstract. Includes bibliographical references (p. 86-87).