Finite difference

About: Finite difference is a research topic. Over the lifetime, 19693 publications have been published within this topic receiving 408603 citations.

Finite difference modeling of Biot's poroelastic equations at seismic frequencies

Abstract: [1] Across the seismic band of frequencies (loosely defined as <10 kHz), a seismic wave propagating through a porous material will create flow in the pore space that is laminar; that is, in this low-frequency ‘‘seismic limit,’’ the development of viscous boundary layers in the pores need not be modeled. An explicit time stepping staggered-grid finite difference scheme is presented for solving Biot’s equations of poroelasticity in this low-frequency limit. A key part of this work is the establishment of rigorous stability conditions. It is demonstrated that over a wide range of porous material properties typical of sedimentary rock and despite the presence of fluid pressure diffusion (Biot slow waves), the usual Courant condition governs the stability as if the problem involved purely elastic waves. The accuracy of the method is demonstrated by comparing to exact analytical solutions for both fast compressional waves and slow waves. Additional numerical modeling examples are also presented.

Operator splitting methods for pricing American options under stochastic volatility

Abstract: We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.

Laminar-flow heat-transfer in non-circular ducts.

Abstract: A numerical method is employed to obtain solutions for laminar flow heat transfer with fully developed velocity profiles and invariant fluid physical properties for rectangular ducts of various aspect ratios with the thermal boundary conditions of constant wall temperature and constant heat input per unit length of the duct. Since an analytical solution for the fully developed velocity profile in a rectangular duct is available, the varying temperature profile remains to be solved numerically from the energy equation which is transformed into a finite difference form by means of two finite difference operators in two dimensions. Numerical values of the initial and boundary temperatures are fixed by choosing a suitable dimensionless temperature depending upon the, thermal boundary condition. As computation involved is very lengthy, a fast digital computer is required. Numerical results obtained from an I.C.T. Atlas computer are presented as the variation of the Nusselt number with the Graetz number. The numerical method is extended to analyse heat transfer with simultaneously aeveloping velocity and temperature profiles. To determine the development of the velocity profile, some simplifications of the Navier-Stokes equation are made. Results are presented for various aspect ratios with the Prandtl number of 0.72. The effect of Prandtl number on heat transfer is also illustrated by numerical results. The numerical method is also used to solve for heat transfer in right-angled isosceles and equilateral triangular ducts with the same hydraulic and thermal boundary conditions as in the previous cases. The predicted results are compared with experimental data. For constant wall temperature, they agree well for Graetz numbers under 70; for constant heat input per unit length, closer agreement is shown over a much wider range of the Graetz numbers. Accuracy of the numerical method is confirmed by the facts that variations of the predicted Nusselt numbers obtained here follow the same trends as those for circular ducts and parallel plates and at the Graetz number of zero, they approach values of the limiting Nusselt numbers obtained by other methods.

Finite difference solutions of solidification phase change problems: Transformed versus fixed grids

Abstract: In using finite difference techniques for solving diffusion/ convection controlled solidification processes, the numerical discretization is commonly carried out in one of two ways: (1) transformed grid, in which case the physical space is transformed into a solution space that can be discretized with a fixed grid in space; (2) fixed grid, in which case the physical space is discretized with a fixed uniform orthogonal grid and the effects of the phase change are accounted for on the definition of suitable source terms. In this paper, recently proposed transformed- and fixed-grid methods are outlined. The two methods are evaluated based on solving a problem involving the melting of gallium. Comparisons are made between the predictive power of the two methods to resolve the position of the moving phase-change front