| This thesis introduces a new fourth order finite difference approximation for the second derivative. We show that this fourth order discretization of the second derivative, which uses the function and its first derivative, is coercive in the discrete inner product space, much like the usual central difference method. This will be useful later in showing the convergence of high order compact (HOC) finite difference methods based on this scheme to solutions of parabolic equations. Finite difference methods of this type fall under the category of Hermitian methods, which were introduced by Collatz [8], and are inspired by the so called box schemes introduced by H. Keller [19]. We start by applying our method to the simple heat equation, examining the stability properties of an explicit finite difference scheme. Next comes the more general linear convection-diffusion equation, where we analyze the stability and convergence properties of a number of variants of our method. We also give an implementation of the scheme, illustrating the combination of the discretized equation with the constitutive Hermitian relation. Numerical examples, and comparisons with the standard implicit and Crank Nicolson methods follow.; Next, we move on to nonlinear parabolic equations, where we review some recent theoretical results about the existence and "blowup" of certain solutions and their derivatives. We adapt our OBIS method to the nonlinear convection term, and prove it converges. We then verify these results with our method, and obtain precise results for the first derivative until its "blowup".; In the two dimensional case, we obtain an approximate factorization of the partial differential equation, which preserves the constitutive relation exactly. The order of the scheme is thus preserved in the two dimensional case, while the computational efficiency associated with ADI schemes is achieved. This factorization also gives an elegant way of dealing with mixed second derivatives.; Some of the applications motivating this work come from the realm of financial mathematics, where high order finite difference schemes can be quite useful. A recently proposed application by During et al. [11] of a fourth order compact scheme analyzed by Rigal in [27] successfully simulates the transaction-cost non-linear model of Barles and Soner. Two significant advantages of the method proposed here are that the coefficients of the stencil are independent of the equation, and the fourth order approximation for the spatial derivative is a "free byproduct". In the modeling of options, this implies a highly accurate hedge ratio at all grid-points. Other one-dimensional nonlinear models where a high order scheme is useful include the transaction-cost model of Avellaneda and Paras [1] and the feedback model of Frey [14]. Besides one dimensional nonlinear models, efficient fourth order schemes are useful in higher dimensional problems, such as the widely used stochastic volatility model of Heston [16]. |
