Finite difference

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

A numerical study of two-dimensional spontaneous rupture propagation

Abstract: Summary We present a numerical technique to determine the displacement and stress fields due to propagation of two-dimensional shear cracks in an infinite, homogeneous medium which is linearly elastic everywhere off the crack plane. Starting from the representation theorem, an integral equation for the displacements inside the crack is found. This integral equation is solved by a method proposed by Hamano for various initial and boundary conditions on the crack surface. We verified the accuracy of our numerical method by comparing it with the analytical solution of Kostrov, and the numerical solution of Madariaga. A critical stress jump across the tip of a crack (between a grid-point inside the crack and a neighbouring point out-side the crack) is used as our fracture criterion. We find that our critical stress jump is the finite difference approximation to the critical stress-intensity factor used in Irwin's fracture criterion. For an in-plane shear crack starting from the Griffin critical length and controlled by the above fracture criterion, the propagation velocity of the crack-tip is found to be sub-Rayleigh or super-shear depending on the strength of the material (given by the critical stress jump) and the instantaneous length of the crack. In fact, the crack-tip velocity may even reach the P-wave velocity for low-strength materials. Additionally we find that once the crack starts propagating, it accelerates rapidly to its terminal velocity, and that the average rupture velocity over an entire length of fault cannot be much smaller than the terminal velocity, for smooth rupture propagation.

Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as Cell-Centered Finite Differences

Abstract: We present an expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order Raviart--Thomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cell-centered finite difference method requiring the solution of a sparse, positive semidefinite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a 9-point stencil in two dimensions and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order approximations in the $L^2$- and $H^{-s}$-norms (and superconvergence is obtained between the $L^2$-projection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If $h$ denotes the maximal mesh spacing, then the optimal rate is $O(h)$. The superconvergence rate $O(h^{2})$ is obtained for the scalar unknown and rate $O(h^{3/2})$ for its gradient and flux in certain discrete norms; moreover, the full $O(h^{2})$ is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.

Split-step methods for the solution of the nonlinear Schro¨dinger equation

Abstract: A split-step method is used to discretize the time variable for the numerical solution of the nonlinear Schrodinger equation. The space variable is discretized by means of a finite difference and a Fourier method. These methods are analyzed with respect to various physical properties represented in the NLS. In particular it is shown how a conservation law, dispersion and instability are reflected in the numerical schemes. Analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived. These results are demonstrated by numerical examples.

Applied Mathematics and Modeling for Chemical Engineers

Abstract: Formulation of Physicochemical Problems. Solution Techniques for Models Yielding Ordinary Differential Equations (ODE). Series Solution Methods and Special Functions. Integral Functions. Staged-Process Models: The Calculus of Finite Differences. Approximate Solution Methods for ODE: Perturbation Methods. Numerical Solution Methods (Initial Value Problems). Approximate Methods for Boundary Value Problems: Weighted Residuals. Introduction to Complex Variables and Laplace Transforms. Solution Techniques for Models Producing PDEs. Transform Methods for Linear PDEs. Approximate and Numerical Solution Methods for PDEs. Appendices. Nomenclature. Postface. Index.