Finite difference

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

A Two-Grid Finite Difference Scheme for Nonlinear Parabolic Equations

Abstract: We present a two-level finite difference scheme for the approximation of nonlinear parabolic equations. Discrete inner products and the lowest-order Raviart--Thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme. Analysis of the scheme is given assuming an implicit time discretization. In this two-level scheme, the full nonlinear problem is solved on a "coarse" grid of size H. The nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points. The resulting linear but nonsymmetric system is solved on a "fine" grid of size h. Some a priori error estimates are derived which show that the discrete L\infty(L2) and L2(H1) errors are $O(h^2 + H^{4-d/2} + \Delta t)$, where $d \geq 1$ is the spatial dimension.

Mass conservative numerical solutions of the head‐based Richards equation

Abstract: Numerical procedures for efficient mass conservative solutions of the head-based form of the Richards equation are presented. Mass conservative solutions are shown to result when the capacity coefficient, C, is formulated by equating the storage term and its chain rule expansion in their discretized forms. Equivalence in the storage term expansion is maintained in finite difference models when C is evaluated with a standard chord slope approximation. This scheme is shown to produce excellent global mass balance accuracy in simulations of vertical moisture infiltration. An analogous approach to the expansion of the storage term using finite elements results in element dependent expressions of C. Application of this approach produces mass balance accuracy with errors less than 1%, but also exhibits slow convergence in the consistent form. A nontraditional finite element procedure is presented which maintains equivalence in the storage term expansion when C is evaluated with the standard chord slope approximation. This scheme exhibits excellent mass balance accuracy, in either the consistent or lumped forms, without significant loss in computational efficiency.

A fourth-order accurate finite-difference scheme for the computation of elastic waves

Abstract: A finite difference for elastic waves is introduced. The model is based on the first order system of equations for the velocities and stresses. The differencing is fourth order accurate on the spatial derivatives and second order accurate in time. The model is tested on a series of examples including the Lamb problem, scattering from plane interf aces and scattering from a fluid-elastic interface. The scheme is shown to be effective for these problems. The accuracy and stability is insensitive to the Poisson ratio. For the class of problems considered here it is found that the fourth order scheme requires for two-thirds to one-half the resolution of a typical second order scheme to give comparable accuracy.

High-frequency response of atomic-force microscope cantilevers

Abstract: Recent advances in atomic-force microscopy have moved beyond the original quasistatic implementation into a fully dynamic regime in which the atomic-force microscope cantilever is in contact with an insonified sample. The resulting dynamical system is complex and highly nonlinear. Simplification of this problem is often realized by modeling the cantilever as a one degree of freedom system. This type of first-mode approximation (FMA), or point-mass model, has been successful in advancing material property measurement techniques. The limits and validity of such an approximation have not, however, been fully addressed. In this article, the complete flexural beam equation is examined and compared directly with the FMA using both linear and nonlinear examples. These comparisons are made using analytical and finite difference numerical techniques. The two systems are shown to have differences in drive-point impedance and are influenced differently by the interaction damping. It is shown that the higher modes must be included for excitations above the first resonance if both the low and high frequency dynamics are to be modeled accurately.