A highly accurate compact finite difference method and its applications in financial mathematics and computational biology

This thesis is concerned with the symbolic generation of finite difference schemes, especially so-called compact finite difference schemes, and their numerical applications to second-order elliptic equations, integro-differential equations, the American option pricing problem in financial mathematics, and cardiac tissue models.; We take as a base Corless and Rokicki's 1995 work on automatic generation of finite difference formulae and numerical integration formulae of univariate and bivariate problems. We then extend this methodology to any dimension. The new Maple routine FINDIF includes higher dimensional cases. It allows for automatic, symbolic discretization of various finite difference formulae, integration formulae, and computes formulae for truncation errors. The Maple implementation is shown to be useful and capable of solving a wide range of problems.; A fourth-order compact finite difference scheme is given for general forms of two point boundary value problems and two dimensional elliptic partial differential equations (PDE's) in simple geometries. By decomposing the coefficient matrix into a sum of several matrices, we prove that the compact finite difference scheme converges with fourth order accuracy. To solve the discretized block tri-diagonal matrix equations arising from the two dimensional elliptic PDE's, we use an efficient iterative method, namely the full multi-grid method. The results show that the compact finite difference scheme is a highly efficient and accurate method.; Compact finite difference methods are applied to integro-differential equations (IDE's) with different boundary conditions, and to a system of IDE's. Both error estimates and numerical experiments confirm that compact finite difference methods can get high-order accuracy. The algorithm can also be used to solve nonlinear IDE's and unsplit-kernel IDE's.; The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation with initial values. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. We show three ways of combining a compact finite difference method for American option price with methods for dealing with this optimal exercise boundary. Through comparison with existing popular methods by numerical experiments, we show that compact finite difference methods provide an exciting new tool for American option pricing.; Finally, a highly efficient, accurate and unconditionally stable algorithm is developed to solve a partial differential equation for simulating the action potential propagation through two dimensional cardiac tissues. In the new algorithm for the two dimensional Luo-Rudy "phase I" action potential model, we discretize the space domain by combining a compact finite difference scheme with an alternating direction implicit (ADI) scheme, which has fourth-order accuracy for interior mesh points, and second-order accuracy for boundary mesh points. A pseudo-ECG is used to study the properties of spiral wave propagation. The new algorithm can be extended to three-dimensional cases.; Keywords. automatic generation; compact finite difference method; convergence; finite difference method, finite element method, American option; integro-differential equation; multi-grid method; elliptic equation; smallest eigenvalue; high accuracy; alternating direction implicit scheme; ADI; Luo-Rudy phase.