Numerical Solution of Boundary Value Problems in Computational Finance

When solving partial differential equations and their boundary conditions withnumerical methods there are three important issues to consider: 1. The type of problem and method. Three general methods are available: Finite difference, Collocation and Finite Element methods. Each have their advantages and disadvantages which are described in this article. 2. The specific method. Having chosen the general method, the specifics of the method must be determined to obtain a solution which is of acceptable precision to the user. This step requires some knowledge of convergence and error estimation theory which is adressed in this article. 3. The programming environment. Today the user has the choice between many diverse programming environments among which are the standard, fast, compiled languages like C/C++ and Fortran, the symbolic, interpreted environments like Maple and Mathematica, the internet directed Java etc. Some of the many possibilities including the possibility of using existing software packages is discussed in this article.