Finite difference method

Abstract: A three‐dimensional finite‐difference (FD) method is used to simulate sonic wave propagation in a borehole with an inhomogeneous solid formation. The second‐order FD scheme solves the first‐order elastic wave equations with central differencing in both space and time via staggered grids. Liao’s boundary condition is used to reduce artificial reflections from the finite computational domain. In the staggered grids, sources have to be implemented at the discrete center in order to maintain the appropriate symmetry in an axisymmetric borehole environment. The FD scheme is validated for multipole sources in three special media: (i) a homogeneous medium; (ii) a homogeneous formation with a fluid‐filled borehole; and (iii) a horizontally layered formation. The staircase approximation of a circular borehole introduces little error in dipole wave fields, although it causes a noticeable phase velocity error in the monopole Stoneley wave. This error has been drastically reduced by using a material averaging scheme ...

Abstract: Numerical solutions of the Benjamin-Bona-Mahony-Burgers equation in one space dimension are considered using Crank-Nicolson-type finite difference method. Existence of solutions is shown by using the Brower's fixed point theorem. The stability and uniqueness of the corresponding methods are proved by the means of the discrete energy method. The convergence in L∞-norm of the difference solution is obtained. A conservative difference scheme is presented for the Benjamin-Bona-Mahony equation. Some numerical experiments have been conducted in order to validate the theoretical results.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

Abstract: This article presents a reduced basis method for constructing a reduced order system for control problems governed by nonlinear partial differential equations. The major advantage of the reduced basis method over others based on finite element, finite difference or spectral method is that it may capture the essential property of solutions with very few basis elements. The feasibility of this method is demonstrated for boundary control problems modeled by the incompressible Navier-Stokes and related equations with the boundary temperature control and boundary electromagnetic control in channel flows.