Numerical solution of differential equations with applications to computational finance and numerical weather forecasting

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I am engaged in several research projects to develop, analyze, test and evaluate numerical methods and to construct robust mathematical software.  I hope these projects will ultimately lead to improvements in mathematical software, enabling scientists, engineers and financial modelers either to solve previously intractable problems or to solve problems more efficiently and/or reliably than is currently possible. Previously, my research focused primarily on initial-value problems (IVPs) for ordinary differential equations (ODEs), but it has since broadened to include boundary value problems (BVPs) for ODEs as well as the application of ODE techniques to partial differential equations (PDEs) and to the solution of practical problems in science, engineering and finance.  I have a secondary interest in numerical linear algebra, particularly problems arising from differential equations.  Over the next few years, I will concentrate on the following two major projects: (1) the study of guaranteed error bounds for the numerical solution of IVPs for ODEs, with the long term goal of developing a robust, reliable, easy-to-use package incorporating these schemes; (2) the study of numerical methods for finance, with particular emphasis on effective computational techniques for computing the price of convertible bonds and the price of credit derivatives.