The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasi-continuum method.

The application of the perfectly matched layer in numerical

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an e cient tool for simulating elastic wave propagation in realistic geological structures in two- and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, de ned with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss–Lobatto–Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an e cient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor=multicorrector format. Long term energy conservation and stability properties are illustrated as well as the e ciency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface di raction is obtained at a low computational cost. The method is shown to provide

. A family of Galerkin finite element methods is presented to accurately and efficiently solve the wave equation that includes sharp propagating wave fronts. The new methodology involves different finite element discretizations at different time levels; thus, at any time level, relatively coarse grids can be applied in regions where the solution changes smoothly while finer grids can be employed near wave fronts. The change of grid from time step to time step need not be continuous, and the number of grid points at different time levels can be arbitrarily different. The formulation is applicable to general second order hyperbolic equations. Stability results are proved and apriori error estimates are established for several boundary conditions. Our error estimates consist of three parts: the time finite difference discretization error, the spatial finite element discretization error, and the error due to the projections of the approximated solution from old grids onto new grids. 1. INT...

This article introduces a new fourth-order implicit time-stepping scheme for the numerical solution of the acoustic wave equation, as a variant of the conventional modified equation method. For an efficient simulation, the scheme incorporates a locally one-dimensional (LOD) procedure having the splitting error of O(At4). Its stability and accuracy are compared with those of the standard explicit fourth-order scheme. It has been observed from various ex- periments for 2D problems that (a) the computational cost of the implicit LOD algorithm is only about 40% higher than that of the explicit method, for the problems of the same size, (b) the implicit LOD method produces slightly less dispersive solutions in heterogeneous media, and (c) its numerical stability and accuracy match well with those of the explicit method.

This article is concerned with accurate and ecient numerical methods for solving viscous and nonviscous wave equations. A three-level second-order implicit algorithm is considered without introducing auxiliary variables. As a perturbation of the algorithm, a locally one-dimensional (LOD) procedure which has a splitting error not larger than the truncation error is suggested to solve problems of diagonal diusion tensors in cubic domains eciently. Both the three-level algorithm and its LOD procedure are proved to be unconditionally stable. An error analysis is provided for the numerical solution of viscous waves. Numerical results are presented to show the accuracy and eciency of the new algorithms for the propagation of acoustic waves and of microscale heat transfer.

Finite difference methods are becoming increasingly popular for calculating synthetic seismograms because of the ease with which they can be applied to model the low-frequency response of complex geometries for which no analytical solutions can be derived. In addition to the obvious advances in computing speed and storage capabilities which have made numerical solution of large, realistic geometries possible, Clayton and Engquists ' (1977) development of absorbing boundaries for acoustic and elastic wave equations greatly reduced the physical storage and com-putational burden necessary to solve a given problem. In the past, most difference calculations used second-order temporal and spatial difference operators (Boore, 1970; and many others). Recently, more attention has been given to higher order spatial difference operators as a means of improving the bandlimiting criteria required to minimize grid dispersion (Alford et al., 1974; Frankel and Clayton, 1984). Unfortunately, no clear guidelines have emerged in the literature for developing absorbing boundary conditions for higher order operators. (In the following, I will refer to second- or fourth-order operators and equations,

The split-step Fourier method is used here to prestack migrate two synthetic borehole-to-surface shot gathers. Model structures in the zone of specular illumination beneath the shot are reconstructed by using the split-step Fourier method both to back-propagate the recorded wave-field and to forward propagate the source wavelet. The overburden is vertically and laterally inhomogeneous. Each depth interval is treated as a homogeneous strip with the mean velocity plus an inhomogeneity correction term. The inhomogeneity correction term is split and spatially multiplied with each spectral component of the wave-field on its entry to and upon its exit from each strip. Propagation through each strip is effected by multiplication in the spatial frequency domain. The split-step Fourier method offers a valuable alternative to finite-difference migration for machines with limited memory. Three imaging methods are compared for two signal-to-noise ratios. They are: image extract...

In this paper we develop a method for the simulation of wave propagation on artificially bounded domains. The acoustic wave equation is solved at all points away from the boundaries by a pseudospectral Chebychev method. Absorption at the boundaries is obtained by applying one-way wave equations at the boundaries, without the use of damping layers. The theoretical reflection coefficient for the method is compared to theoretical estimates of reflection coefficients for a Fourier model of the problem. These estimates are confirmed by numerical results. Modification of the method by a transformation of the grid to allow for better resolution at the center of the grid reduces the maximum eigenvalues of the differential operator. Consequently, for stability the maximum timestep is O(1=N) as compared to O(1=N 2 ) for the standard Chebychev method. Therefore, the Chebychev method can be implemented with efficiency comparable to that of the Fourier method. Moreover, numerical results presente...

. A naturally parallelizable formulation is considered for solving linearized time-dependent Navier-Stokes equations. The evolution problem is first converted into a complex valued elliptic system by Fourier transformation. Existence and uniqueness are then given for the resulting problems for each frequency. Stability and regularity depending on frequency are analyzed. Next, standard finite element methods are used to approximate solutions for the transformed elliptic systems. Finally, time-dependent solutions are constructed by Fourier inversion with a full estimate of errors generated in the truncation in the Fourier transformation, quadrature rules, and finite element approximations. 1. Introduction The domain\Omega will be assumed to be a bounded Lipschitz domain in R N ; N = 2; 3; with the boundary \Gamma. Set J = (0; T ). We consider the following linearized Navier-Stokes equations: @u @t \Gamma ¯\Deltau + (U \Delta r)u +rp = f; \Omega \Theta J; (1.1a) r \Delta u = 0; \...