Finite difference method

About: Finite difference method is a research topic. Over the lifetime, 21603 publications have been published within this topic receiving 468852 citations. The topic is also known as: Finite-difference methods & FDM.

A novel fitted finite volume method for the Black-Scholes equation governing option pricing

Abstract: In this paper we present a novel numerical method for a degenerate partial differential equation, called the Black-Scholes equation, governing option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. To derive the error bounds for the spatial discretization of the method, we formulate it as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. Stability of the discretization is proved and an error bound for the spatial discretization is established. It is also shown that the system matrix of the discretization is an M-matrix so that the discrete maximum principle is satisfied by the discretization. Numerical experiments are performed to demonstrate the effectiveness of the method.

Direct Numerical Simulations of Flow Past an Array of Distributed Roughness Elements

Abstract: Direct numerical simulation was used to describe the subsonic flow past an array of distributed cylindrical roughness elements mounted on a flat plate. Solutions were obtained for element heights corresponding to a roughness-based Reynolds number (Re k ) of both 202 and 334. The numerical method used a sixth-order-accurate centered compact finite difference scheme to represent spatial derivatives, which was used in conjunction with a tenth-order low-pass Pade-type nondispersive filter operator to maintain stability. An implicit approximately factored time-marching algorithm was employed, and Newton-like subiterations were applied to achieve second-order temporal accuracy. Calculations were carried out on a massively parallel computing platform, using domain decomposition to distribute subzones on individual processors. A high-order overset grid approach preserved spatial accuracy on the mesh system used to represent the roughness elements. Features of the flowfields are described, and results of the computations are compared with experimentally measured velocity components of the time-mean flowfield, which are available only for Re k = 202. Flow about the elements is characterized by a system of two weak corotating horseshoe vortices. For Re k = 334, an unstable shear layer emanating from the top of the cylindrical element generated nonlinear unsteady disturbances of sufficient amplitude to produce explosive bypass transition downstream of the array. The Re k = 202 case displayed exponential growth of turbulence energy in the streamwise direction, which may eventually result in transition.

Mimetic Finite Difference Methods for Diffusion Equations

Abstract: This paper reviews and extends the theory and application of mimetic finite difference methods for the solution of diffusion problems in strongly heterogeneous anisotropic materials. These difference operators satisfy the fundamental identities, conservation laws and theorems of vector and tensor calculus on nonorthogonal, nonsmooth, structured and unstructured computational grids. We provide explicit approximations for equations in two dimensions with discontinuous anisotropic diffusion tensors. We mention the similarities and differences between the new methods and mixed finite element or hybrid mixed finite element methods.

Finite-difference modelling of magnetotelluric fields in two-dimensional anisotropic media

Abstract: SUMMARY An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field, Ex and H,. These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.