Finite-difference Problem Research Articles

Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences

Abstract This article provides an overview of the application of the staggered-grid finite-difference technique to model wave propagation problems in 3D elastic media. In addition to presenting generalized, discrete representations of the differential equations of motion using the staggered-grid approach, we also provide detailed formulations that describe the incorporation of moment-tensor sources, the implementation of a stable and accurate representation of a planar free-surface boundary for 3D models, and the derivation and implementation of an approximate technique to model spatially variable anelastic attenuation within time-domain finite-difference computations. The comparison of results obtained using the staggered-grid technique with those obtained using a frequency-wavenumber algorithm shows excellent agreement between the two methods for a variety of models. In addition, this article also introduces a memory optimization procedure that allows large-scale 3D finite-difference problems to be computed on a conventional, single-processor desktop workstation. With this technique, model storage is accommodated using both external (hard-disk) and internal (core) memory. To reduce system overhead, a cascaded time update procedure is utilized to maximize the number of computations performed between I/O operations. This formulation greatly expands the applicability of the 3D finite-difference technique by providing an efficient and practical algorithm for implementation on commonly available workstation platforms.

An Incomplete Domain Decomposition Preconditioning Method for Nonlinear Nodal Kinetics Calculations

Methods are proposed for the efficient parallel solution of nonlinear nodal kinetics equations. Because the two-node calculation in the nonlinear nodal method is naturally parallelizable, the majority of the effort is devoted to the development of parallel methods for solving the coarse-mesh finite difference (CMFD) problem. A preconditioned Krylov subspace method (biconjugate gradient stabilized) is chosen as the iterative algorithm for the CMFD problem, and an efficient parallel preconditioning scheme is developed based on domain decomposition techniques. An incomplete lowerupper triangular factorization method is first formulated for the coefficient matrices representing each three-dimensional subdomain, and coupling between subdomains is then approximated by incorporating only the effect of the nonleakage terms of neighboring subdomains. The methods are applied to fixed-source problems created from the International Atomic Energy Agency three-dimensional benchmark problem. The effectiveness of the incomplete domain decomposition preconditioning on a multiprocessor is evidenced by the small increase in the number of iterations as the number of subdomains increases. Through the application to both CMFD-only and nodal calculations, it is demonstrated that speedups as large as 49 with 96 processors are attainable in the nonlinear nodal kinetics calculations.

Molecular dynamics simulation with a continuum electrostatic model of the solvent

AbstractThe accuracy and simplicity of the Poisson‐Boltzmann electrostatics model has led to the suggestion that it might offer an efficient solvent model for use in molecular mechanics calculations on biomolecules. We report a successful merger of the Poisson‐Boltzmann and molecular dynamics approaches, with illustrative calculations on the small solutes dichloroethane and alanine dipeptide. The algorithm is implemented within the program UHBD. Computational efficiency is achieved by the use of rather coarse finite difference grids to solve the Poisson‐Boltzmann equation. Nonetheless, the conformational distributions generated by the new method agree well with reference distributions obtained as Boltzmann distributions from energies computed with fine finite difference grids. The conformational distributions also agree well with the results of experimental measurements and conformational analyses using more detailed solvent models. We project that when multigrid methods are used to solve the finite difference problem and the algorithm is implemented on a vector supercomputer, the computation of solvent electrostatic forces for a protein of modest size will add only about 0.1 s computer time per simulation step relative to a vacuum calculation. © 1995 by John Wiley & Sons, Inc.

The parallel computation of solutions to electrostatic problems using multigrid techniques

AbstractThe application of multigrid techniques to the computation of the static solutions of electromagnetic field problems governed by Laplace's equation is described. This technique is compared with the conventional successive over‐relaxation (SOR) method for solving finite difference problems. In contrast to SOR, the number of iterations of multigrid needed to achieve convergence is largely independent of the grid size. It is shown that the relative performance of multigrid is excellent on large grids where the number of iterations of SOR needed to achieve convergence becomes prohibitively large. The technique is illustrated by applying a parallel implementation of multigrid to find a quasistatic solution of a boxed microstrip problem.

Convergence of approximate solutions of a differential equation

(1.2) u(0, x) = g(x), x ∈ R , where a is continuous and g is a bounded and measurable function. The solution of problem (1.1)–(1.2) is investigated in Section 2 (Theorem 1). Section 3 contains the definition of the finite-difference problem, in Section 4 the problem approximating (1.1)–(1.2) is formulated, and the theorems concerning the convergence of the numerical solution are stated (Theorem 2 for g ∈W 2 1 (R), Theorem 3 for g ∈ L1(R)). Section 5 contains some results of numerical computation. In the next sections all the results are proved. Let us now define some function spaces which will be used in the paper. First, the spaces Lp are defined in the usual way, and we use the following norms and moduli of continuity: if f ∈ Lp(A), A ⊂ R, then ‖f‖p = ( ∫

Coupled extensional and bending motion in elastic waveguides

A variety of different linear elastic waves propagate in long slender rods. These waves do not interact in straight rods; however, extensional and bending waves do interact in slightly curved rods. If curved rods are used as waveguides for extensional waves, wave energy is exchanged between extensional and beding waves. The result has the superficial appearance of viscous attenuation. The petroleum and geothermal industries have an interest in using drill pipe as a waveguide to transmit data by acoustic waves. Attempts have been made to transmit data from transducers in the vivinity of a drill bit to the drill rig at the surface. These efforts have not been successful. Poor signal strenght is a major problem. Historically,the observed reduction in signal strength is attributed to viscous dissipation. This paper proposes an alternative model-mode conversion from extensional waves to bending waves. This model is analyzes both as an eigenvalue problem and a transient finite-difference problem. The results predict measured attenuation levels, and account for many anomalous observations.

Development of Generic Field Classes for Finite Element and Finite Difference Problems

This article considers the development of a reusable object‐oriented array library, as well as the use of this library in the construction of finite difference and finite element codes. The classes in this array library are also generic enough to be used to construct other classes specific to finite difference and finite element methods. We demonstrate the usefulness of this library by inserting it into two existing object‐oriented scientific codes developed at Sandia National Laboratories. One of these codes is based on finite difference methods, whereas the other is based on finite element methods. Previously, these codes were separately maintained across a variety of sequential and parallel computing platforms. The use of object‐oriented programming allows both codes to make use of common base classes. This offers a number of advantages related to optimization and portability. Optimization efforts, particularly important in large scientific codes, can be focused on a single library. Furthermore, by encapsulating machine dependencies within this library, the optimization of both codes on different architec‐tures will only involve modification to a single library.

Ordering Methods for Preconditioned Conjugate Gradient Methods Applied to Unstructured Grid Problems

It is well known that the ordering of the unknowns can have a significant effect on the convergence of preconditioned conjugate gradient (PCG) methods. There has been considerable experimental work on the effects of ordering for finite difference problems. In many cases, good results have been obtained with preconditioners based on diagonal, spiral, red / black reduced system orderings, or some others. The reduced system approach generally gives rapid convergence. There has been comparatively less work on the effect of ordering for finite element problems on unstructured meshes. In this paper, an ordering technique for unstructured grid problems is developed. At any stage of the partial elimination, the next pivot node is selected so as to minimize the norm of the discarded fill matrix. Numerical results are given for model problems and for problems arising in groundwater contamination. Computations are reported for two-dimensional triangular grids, and for three-dimensional tetrahedral grids. The example...

A 3D spectral multigrid method

We consider a 3D spectral multigrid method for 3D elliptic problems with arbitrary anisotropies. The smoothing method is a preconditioned Richardson iteration, where a pseudospectral discretization is used on the right-hand side, and a finite-difference discretization on the left-hand side. In each step of this iteration, a finite-difference problem is solved approximately, by plane-relaxation sweeps. The plane relaxation requires the solution of 2D finite-difference problems. These problems are solved approximately using 2D multigrid cycles.

Multigrid methods for combined finite difference and fourier problems

Multigrid methods for combined finite difference and Fourier pseudospectral problems are considered. Applications to two- and three-dimensional elliptic systems are given. Suitable relaxation schemes for isotropic and anisotropic problems are presented. Numerical results demonstrate the efficiency of these methods.

A multipoint finite-difference scheme for the problem of bending of rectangular orthotropic plates with freely supported edges: Construction and convergence estimate

The boundary-value problem $$\begin{gathered} D_1 \frac{{\partial ^4 w}}{{\partial x^4 }} + 2D_2 \frac{{\partial ^4 w}}{{\partial x^2 \partial y^2 }} + D_3 \frac{{\partial ^4 w}}{{\partial y^4 }} = f\left( {x,y} \right) \hfill \\ w\left| {_{y = 0;b} } \right. = 0, \frac{{\partial ^2 w}}{{\partial y^2 }}\left| {_{y = 0;b} } \right. = 0; w\left| {_{x = - a;a} } \right. = 0,\frac{{\partial ^2 w}}{{\partial y^2 }}\left| {_{x = - a;a} } \right. = 0 \hfill \\ \end{gathered} $$ of static deflection of a rectangular orthotropic plate is replaced with a finite-difference problem. The rectangle [−a⩽x⩽a, O⩽y⩽b] is partitioned into a mesh with step h in the direction y and h1, in the direction x; second derivatives with respect to y and x are replaced with multipoint approximations using the templates 2p+1, 2p1+1 (where p and p1 · are arbitrary natural numbers) with errors O(h2p), O(h2p1); the fourth-order derivatives are replaced with approximations using the templates 4p+1 and 4p1+1 with the same errors. The finite-difference system of linear algebraic equations is transformed into a decomposable system. The convergence of the proposed method is estimated.

The solution of linear equations on a SIMD computer using a parallel iterative algorithm

Techniques for solving linear equations on a single instruction multiple data (SIMD) computer such as the ICL DAP have so far been confined to simple methods such as the Successive Overrelaxation and Alternating Direction Implicit algorithms. While these techniques are adequate for simple finite difference problems require more complex algorithms. Preconditioned conjugate gradient methods have solved difficult problems successfully on serial machines. This paper describes a preconditioning technique suitable for parallel machines and numerical results obtained from a series of problems of varying degrees of difficulty.

Lp-estimates for the solutions of a finite-difference boundary-value problem

Lp-estimates for the solutions of a finite-difference boundary-value problem

The Störmer approach and the solution of two coupled Schrödinger equations

In a recent work (1), MELNIKOV et al. investigated the usefulness of the StSrmer approach {2) for the practical solutions of quantum mechanical scattering problems in terms of the finite-difference analogues of the SchrSdinger equation. This approach seems to be quite unique for formulating the boundary conditions for the corresponding SehrSdinger-StSrmer equation without using starting methods and hence is of principle importance for the solution of a relevant finite-difference inverse problem. In view of its numerical advantage and accuracy, the purpose of this note is to incorporate the StSrmer technique in case of two coupled SchrSdinger equations by separating them through an analytic procedure into two equivalent (uncoupled) equations. As well known, a system of two coupled SchrSdinger equations is a very convenient ground for analysing the relative convergence of several methods for calculating the observable characteristics of the elastic and inelastic scattering problem through the comparison with corresponding analytically solvable problems (a). Unfortunately there are very few cases in which au analytical solution is available and even there it is necessary to observe severe restrictions on the form of the functions and limits of the variables involved (4). Consequently, the influence of related physical phenomena -en te r ing through more complicated forms of, say, potential form factors and the effect of higher partial waves together with the internal structure of the interacting par t ic les-is masked due to the truncation requirement. At this stage although a direct integration of the coupled set of partial differential equations may be carried out with a good accuracy, the methods involved become cumbersome when the model is sophisticated (5). Furthermore, in some cases the intermediate physical concepts, e.g. the magnitude of variations in the observable quantities due to the coupling of particular channel to the

The Effects of Loading and Self-Attraction on Global Ocean Tides: The Model and the Results of a Numerical Experiment

The computation of global tides, including the effects of loading and self-attraction, is formulated. A uniqueness theorem for the solution of the differential and associated finite-difference Problem is presented. The results of the calculation of semi-diurnal tides (M2 wave) are discussed. The effects of loading and self-attraction do not appear to lead to a major alteration of the spatial structure of the phenomenon being studied, but in several regions of the World Ocean significant changes in tidal character are observed.

An approximate method for solving a boundary value problem in the theory of momentless spherical elastic shells

Solution by the mesh method of a problem in the theory of momentless spherical elastic shells, given in the displacements at the edge of the shell, is discussed. The unique solvability and stability of the initial and difference problems is proved, together with the convergence of the method as the mesh step tends to zero. In this paper we discuss a finite-difference method for solving a boundary value problem for the system of equations of a spherical momentless shell when the displacements on the edge of the shell are given. In Section 1 we show that this boundary value problem is uniquely solvable and stable with respect to the right-hand sides of the system and the edge displacements, i.e. the problem is correctly posed. In Section 2 we discuss the solution of the problem by the mesh method; we describe how the relevant finite-difference problem is obtained, and show that it is uniquely solvable and stable. With regard to the finite-difference solution, we use the stability of the initial problem, proved in Section 1, to show that it is convergent to the exact solution and we estimate the error for a given mesh step.

Asymptotic behaviour of the solutions of the finite-difference cauchy problem

Asymptotic behaviour of the solutions of the finite-difference cauchy problem

On the convergence of a locally one-dimensional scheme for solving the multidimensional equation of heat conduction on non-uniform meshes

WE examine the convergence of a locally one-dimensional scheme (see [1–5]) for solving, in a sequence of non-uniform meshes, the first boundary value problem in an arbitrary region, and the second and third boundary value problems in stepped regions, for the equation of heat conduction containing no mixed derivatives. It will be shown that the schemes considered are convergent in the mesh norm of C in a sequence of non-uniform meshes at the rate O(h 2 ln ( V 0 H ∗) + τ ) where τ is the mesh time-step, V 0 the volume of the region G, and h the maximum step of the space mesh R p h : h = max max ℏ α ( x i ), x iϵ G ̄ 1 ⩽ α ⩽ p where ℏ α ( x i ) is the mean step of R p h at the base-point x i in the direction the coordinate axis ox α , a = 1, 2, …, p, and p is the number of dimensions; H∗ is the minimum cell volume, H ∗= min x iε G H(x i, H(x i= Π α=1 p ℏ α(x i) . The convergence rate in a non-uniform mesh is estimated in the present paper by using both the maximum principle and the method of energy inequalities of the n-th rank developed in [6] (see also [3D; this allows the solution of the finite-difference problem to be estimated in the mesh norm of L 2 n , where n is an arbitrary integer. From this estimate, the convergence in the mesh norm of C is obtained. If only the maximum principle and the associated theorems are used, too low a convergence rate estimate is obtained for the scheme. When obtaining a priori estimates of the n-th rank, the finite-difference operators ΛA α , corresponding to the differential operators ( ∂ ∂x α ) (k α(x, t) ∂/∂x α) (and to homogeneous boundary conditions) would have to be negative in any non-uniform mesh. The operators chosen in [3] do not retain their negative properties for all regions G and meshes R p h . In the present paper, negative operators (on any non-uniform mesh in any region), as introduced in [7], are used for Λ α . Section 1 and 2 will be concerned with constructing, and investigating the convergence of, a locally one-dimensional scheme in a non-uniform mesh in an arbitrary region, in the case of the first boundary value problem. In Section 3 a locally one-dimensional scheme is developed for solving the second and third boundary value problems in stepped regions. These schemes are also uniformly convergent at a rate O(h 2 In ( V 0 H∗ ) + τ) .

Bounds for the Solution of the Sturm-Liouville Problem with Application to Finite Difference Methods

1. Introduction. A common method for obtaining an approximate solution to the various self-adjoint two point boundary value problems for the Sturm-Liouville equation is by finite differences. In defining any finite difference analogue, it is necessary to know that the truncation error (i.e., the difference between the exact solution and the solution of the approximating finite difference problem) can be made arbitrarily small by making the mesh size h small. In addition to knowing the order of convergence, it is desirable to have a priori error bounds of that order to assist one in the selection of h. It is the aim of this paper to show how such an error bound may be obtained for certain finite difference analogues to a class of smooth problems. The point of view taken here is to utilize the existence of a pointwise bound for any sufficiently smooth function y considered. In particular, it can be shown that

Bounds on the truncation error by finite differences for the Goursat problem

the various boundary value problems for partial differential equations is by finite differences. The fundamental problem concerning such finite difference approximations is to show that the truncation error (i.e., the difference between the exact solution and the solution of the approximating finite difference problem) tends to zero with diminishing mesh size. Further the success of a finite difference method is related to the possibility of deducing explicit a priori bounds for the solution, or to the stability of the difference scheme. Various authors have studied the question of stability and a priori estimates connected with finite difference schemes concerning hyperbolic partial differential equations (see e.g. Richtmyer [141, with references). In general the main tool used in obtaining estimates for the solution of hyperbolic equations is the energy method. In this paper we obtain explicit bounds in terms of the data of the problem for