Finite difference

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

On the implementation of perfectly matched layers in a three‐dimensional fourth‐order velocity‐stress finite difference scheme

Abstract: [1] Robust absorbing boundary conditions are central to the utility and advancement of 3-D numerical wave propagation methods. It is in general preferred that an absorbing boundary method be capable of broadband absorption, be efficient in terms of memory and computation time, and be widely stable in connection with sophisticated numerical schemes. Here we discuss these issues for a promising absorbing boundary method, perfectly matched layers (PML), as implemented in the widely used fourth-order accurate three-dimensional (3-D) staggered-grid velocity-stress finite difference (FD) scheme. Numerical results for point (explosive and double couple) and extended sources, velocity structures (homogeneous, 1-D and 3-D), and different thickness PML zones are excellent, in general, leaving no observable reflections in PML seismograms compared to the amplitudes of the primary phases. For both homogeneous half-space and 1-D models, typical amplitude reduction factors (with respect to the maximum trace amplitude) range between 1/100 and 1/625 for PML thicknesses of 5–20 nodes. A PML region of thickness 5 outperforms a simple exponential damping region of thickness 20 in a homogeneous half-space model by a factor of 3. We find that PML is effective across the simulation bandwidth. For example, permanent offset artifacts due to particularly poor absorption of long-period energy by the simple exponential damping are effectively absent when PML is used. The computational efficiency and storage requirements of PML, compared to the simple exponential damping, are reduced due to the need for only narrow absorbing regions. We also discuss stability and present the complete PML model for the 3-D velocity-stress system.

Numerical Approximation of Some Linear Stochastic Partial Differential Equations Driven by Special Additive Noises

Abstract: This paper is concerned with the numerical approximation of some linear stochastic partial differential equations with additive noises. A special representation of the noise is considered, and it is compared with general representations of noises in the infinite dimensional setting. Convergence analysis and error estimates are presented for the numerical solution based on the standard finite difference and finite element methods. The effects of the noises on the accuracy of the approximations are illustrated. Results of the numerical experiments are provided.

Numerical Methods for Evolutionary Differential Equations

Abstract: Methods for the numerical simulation of dynamic mathematical models have been the focus of intensive research for well over 60 years, and the demand for better and more efficient methods has grown as the range of applications has increased. Mathematical models involving evolutionary partial differential equations (PDEs) as well as ordinary differential equations (ODEs) arise in diverse applications such as uid ow, image processing and computer vision, physics-based animation, mechanical systems, relativity, earth sciences, and mathematical nance. This textbook develops, analyzes, and applies numerical methods for evolutionary, or time-dependent, differential problems. Both PDEs and ODEs are discussed from a unified viewpoint. The author emphasizes finite difference and finite volume methods, specifically their principled derivation, stability, accuracy, efficient implementation, and practical performance in various fields of science and engineering. Smooth and nonsmooth solutions for hyperbolic PDEs, parabolic-type PDEs, and initial value ODEs are treated, and a practical introduction to geometric integration methods is included as well. The author bridges theory and practice by developing algorithms, concepts, and analysis from basic principles while discussing efficiency and performance issues and demonstrating methods through examples and case studies from numerous application areas. Audience: This textbook is suitable for researchers and graduate students from a variety of fields including computer science, applied mathematics, physics, earth and ocean sciences, and various engineering disciplines. Gradute students at the beginning or advanced level (depending on the discipline) and researchers in a variety of fields in science and engineering will find this book useful. Researchers who simulate processes that are modeled by evolutionary differential equations will find material on the principles underlying the appropriate method to use and the pitfalls that accompany each method. Contents: Preface; 1 Introduction; 2 Methods and Concepts for ODEs; 3 Finite Difference and Finite Volume Methods; 4 Stability for Constant Coefficient Problems; 5 Variable Coefficient and Nonlinear Problems; 6 Hamiltonian Systems and Long Time Integration; 7 Dispersion and Dissipation; 8 More on Handling Boundary Conditions; 9 Several Space Variables and Splitting Methods; 10 Discontinuities and Almost Discontinuities; 11 Additional Topics; Bibliography; Index.

Methods for the numerical solution of the nonlinear Schroedinger equation

Abstract: Optimal L2 rates of convergence are established for several fully-discrete schemes for the numerical solution of the nonlinear Schroedinger equation. Both finite differences and finite elements are considered for the discretization in space, while the integration in time is treated either by the leap-frog technique or by a modified Crank-Nicolson procedure, which generalizes the one suggested by Delfour, Fortin and Payne and possesses two useful conserved quantities.

Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.