Finite difference method

Abstract: The Pricing Equations. Analysis of Finite Difference Methods. Special Issues. Coordinate Transformations. Numerical Examples. Index.

Abstract: A numerical method for solving the fractional diffusion equation, which could also be easily extended to other fractional partial differential equations, is considered. In this paper we combine the forward time centered space (FTCS) method, well known for the numerical integration of ordinary diffusion equations, with the Grunwald--Letnikov discretization of the Riemann--Liouville derivative to obtain an explicit FTCS scheme for solving the fractional diffusion equation. The stability analysis of this scheme is carried out by means of a powerful and simple new procedure close to the well-known von Neumann method for nonfractional partial differential equations. The analytical stability bounds are in excellent agreement with numerical test. A comparison between exact analytical solutions and numerical predictions is made.

Abstract: A new methodology to build discrete models of boundary-value problems is presented. The h-pcloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomialreproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and a several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater -exibility than traditional h-p finite element methods. Several numerical experiments in I-D and 2-D are also presented. @ 1996 John Wiley & Sons, Inc. In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent years. Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. The analysis of problems such as crack propagation, penetration, and large deformations, can, in principle, be greatly simplified by the use of meshless methods. A growing crack, for example, can be modeled by simply extending the free surfaces that correspond to the crack [ 11. The analysis of large deformation problems by, e.g., finite element methods, may require the continuous remeshing of the domain to avoid the breakdown of the calculation due to

Abstract: A finite difference equation formulation for the equations of elasticity is presented and applied to the problem of a layered half-space with a buried point source emitting a compressional pulse. Complete theoretical seismograms for the horizontal and vertical components of displacement are obtained. The results for a specific case are compared with those found by a completely different method in order to check the validity of the finite difference methods. The agreement is excellent. The effect of different mesh sizes on the theoretical seismograms is studied next and a suitable grid system selected for the applications that follow. The development of Rayleigh waves on the surface of a half-space and the change of the Rayleigh wave with depth and pulse width are examined. The problem of a layered half-space with a high velocity bottom is considered and the refraction arrivals on the surface and on the interface are studied. The problem of interface waves on the surface separating two semiinfinite media is also examined. Interface waves are found when the physical parameters lie both inside and outside the region determined by the Stoneley equation. Finally, a series of theoretical seismograms for a layered half-space showing the variation of the surface waves as a function of depth and of the density in the lower medium is presented.