Finite difference method

Abstract: We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

Abstract: We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Levy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Levy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

Abstract: The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

Abstract: A numerical technique is developed to solve the three-dimensional potential distribution about a point source of current located in or on the surface of a half-space containing arbitrary two-dimensional conductivity distribution. Finite difference equations are obtained for Poisson's equations by using point- as well as area-discretization of the subsurface. Potential distributions at all points in the set defining the half-space are simultaneously obtained for multiple point sources of current injection. The solution is obtained with direct explicit matrix inversion techniques. An empirical mixed boundary condition is used at the “infinitely distant” edges of the lower half-space. Accurate solutions using area-discretization method are obtained with significantly less attendant computational costs than with the relaxation, finite-element, or network solution techniques for models of comparable dimensions.