Finite difference

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

Fully non‐linear free‐surface simulations by a 3D viscous numerical wave tank

Abstract: A finite difference scheme using a modified marker-and-cell (MAC) method is applied to investigate the characteristics of non-linear wave motions and their interactions with a stationary three-dimensional body inside a numerical wave tank (NWT). The Navier-Stokes (NS) equation is solved for two fluid layers, and the boundary values are updated at each time step by a finite difference time marching scheme in the frame of a rectangular co-ordinate system. The viscous stresses and surface tension are neglected in the dynamic free-surface condition, and the fully non-linear kinematic free-surface condition is satisfied by the density function method developed for two fluid layers. The incident waves are generated from the inflow boundary by prescribing a velocity profile resembling flexible flap wavemaker motions, and the outgoing waves are numerically dissipated inside an artificial damping zone located at the end of the tank. The present NS-MAC NWT simulations for a vertical truncated circular cylinder inside a rectangular wave tank are compared with the experimental results of Mercier and Niedzwecki, an independently developed potential-based fully non-linear NWT, and the second-order diffraction computation

Development and Testing of a Pseudo-Three-Dimensional Model of Hydraulic Fracture Geometry

Abstract: A pseudo-three-dimensional (P3DH) model has been developed to describe realistically the evolution of geometry of a 3D hydraulic fracture produced by fluid injection into a reservoir. The present model bypasses the task of a fully 3D crack geometry calculation, which is at present prohibitively complex in the context of a complete practical fracturing simulator. The P3DH model presented formulates the problem in terms of equations for lateral fluid flow and crack opening for the main body of the fracture, coupled with a very efficient scheme for describing vertical fracture growth at each cross section. The equations for lateral flow are solved by finite differences, and the vertical propagation problem is solved by numerical implementation of a singular integral equation on a suitable set of Chebyshev points. The testing of P3DH components shows both an excellent agreement of the lateral propagation model with various analytical solutions and a strong sensitivity of vertical propagation to confining stress and stiffness contrast of adjacent strata and to fluid rheology. The sample simulations show that the model produces realistic fracture growth under a wide range of conditions, is extremely sensitive to the dominant containment parameters, and therefore can be used to study the effect ofmore » relevant design parameters on fracture shapes and pressures in stimulation treatments.« less

A Cyclic Reduction Algorithm for Solving Block Tridiagonal Systems of Arbitrary Dimension

Abstract: A generalization of the Buneman variant of cyclic odd–even reduction algorithm for solving finite difference approximations to Poisson’s equations is presented. This generalization places no restriction on the block size, n, of the system and computes the solution in $O(n^2 \log _2 n)$ operations.