Finite difference method

Abstract: In this article, we construct and analyse a family of finite difference schemes for the acoustic wave equation with variable coefficients. These schemes are fourth-order accurate in space and time in the case of smooth media and are designed to remain stable and "optimal" for reflection-transmission phenomena in the case of discontinuous coefficients. Together with a detailed mathematical study, various numerical results are presented.

Abstract: A new finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations. The present method uses conventional quadrilateral elements for the interpolation of velocity components, so that it can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions). A Picard iteration scheme with either flow rate or elasticity increment is used to treat the non-Newtonian stresses as pseudo-body forces, and an efficient and consistent predictor-corrector scheme is adopted for both the particle-tracking and strain tensor calculations. The new method has been used to simulate entry flows of polymer melts in circular abrupt contractions using the K-BKZ integral constitutive model. Results are in very good agreement with existing numerical data. The important question of mesh refinement and convergence for integral models in complex flow at high flow rate has also been addressed, and satisfactory convergence and mesh-independent results are obtained. In addition, the present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature.

Abstract: We develop a finite difference method to solve partial integro-differential equations which describe the behavior of option prices under jump-diffusion models. With localization to a bounded domain of the spatial variable, these equations are discretized on uniform grid points over a finite domain of time and spatial variables. The proposed method is based on three time levels and leads to linear systems with tridiagonal matrices. In this paper the stability of the proposed method and the second-order convergence rate with respect to a discrete $\ell^{2}$-norm are proved. Numerical results obtained with European put options under the Merton and Kou models show the behaviors of the stability and the second-order convergence rate.

Abstract: Finite-difference acoustic-wave modeling and reverse-time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well. However, conventional finite-difference methods for solving the acoustic- or elastic-wave equation suffer from numerical dispersion when too few samples per wavelength are used. The flux-corrected transport (FCT) algorithm, adapted from hydrodynamics, reduces the numerical dispersion in finite-difference wavefield continuation. The flux-correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion. Incorporating the flux-correction procedure in conventional finite-difference modeling or reverse-time migration can provide finite-difference solutions with no numerical dispersion even for impulsive sources. The FCT correction, which can be applied to finite-difference approximations of any order in space and time, is an efficient alternative to use for finite-difference approximations of increasing order. Through demonstrations of modeling and migration on both synthetic and field data, we show the benefits of the FCT algorithm, as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse.