Abstract: SUMMARY We present a new massively parallel method for computation of first arrival times in arbitrary velocity models. An implementation on conventional sequential computers is also proposed. This method relies on a systematic application of Huygens’ principle in the finite difference approximation. Such an approach explicitly takes into account the existence of different propagation modes (transmitted and diffracted body waves, head waves). Local discontinuities of the time gradient in the first arrival time field (e.g., caustics) are built as intersections of locally independent wavefronts. As a consequence, the proposed method provides accurate first traveltimes in the presence of extremely severe, arbitrarily shaped velocity contrasts. Associated with a simple procedure which accurately traces rays in the obtained time field, this method provides a very fast tool for a large spectrum of seismic and seismological problems. We show moreover that this method may also be used to obtain several arrivals at a given receiver, when the model contains reflectors. This possibility significantly extends the domain of potential geophysical applications.

Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a third-order flux-difference splitting technique for the convective terms and the second-order central difference for the viscous terms. The accuracy is obtained in the numerical solutions by subiterating the equations in pseudotime for each physical time step. The equations are solved with a line-relaxation scheme that allows the use of very large pseudotime steps leading to fast convergence for steady-state problems as well as for the subiterations of time-dependent problems. The steady-state solution of flow through a square duct with a 90-deg bend is computed, and the results are compared with experimental data. Good agreement is observed. Computations of unsteady flow over a circular cylinder are presented and compared to other experimental and computational results. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

Abstract: A rigorous analysis of the numerical error associated with the use of stair-stepped (saw-toothed) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented. First, a dispersion analysis in two dimensions is performed to obtain the numerical reflection coefficient for a plane wave scattered by a perfectly conducting wall, tilted with respect to the axes of the finite-difference grid, under both transverse electric and transverse magnetic polarizations. The characteristic equation for surface waves that can be supported by such saw-tooth conducting surfaces is derived. This equation leads to expressions that show the dependence of the propagation constant along the boundary and the attenuation constant perpendicular to it on cell size and wavelength. Numerical simulations that demonstrate the effects predicted by the dispersion analysis are presented. >

Abstract: We provide a theory for the analysis of multigrid algorithms for symmetric positive definite problems with nonnested spaces and noninherited quadratic forms. By this we mean that the form on the coarser grids need not be related to that on the finest, i.e., we do not stay within the standard variational setting. In this more general setting, we give new estimates corresponding to the \"V cycle, W cycle and a \"V cycle algorithm with a variable number of smoothings on each level. In addition, our algorithms involve the use of nonsymmetric smoothers in a novel way. We apply this theory to various numerical approximations of second-order elliptic boundary value problems. In our first example, we consider certain finite difference multigrid algorithms. In the second example, we consider a finite element multigrid algorithm with nested spaces, which however uses a prolongation operator that does not coincide with the natural subspace imbedding. The third example gives a multigrid algorithm derived from a loosely coupled sequence of approximation grids. Such a loosely coupled grid structure results from the most natural standard finite element application on a domain with curved boundary. The fourth example develops and analyzes a multigrid algorithm for a mixed finite element method using the so-called Raviart-Thomas elements.

Abstract: Field evolution along longitudinally nonuniform finite-cladding fibers of circularly symmetric cross section is analyzed in terms of coupled modes. Local modes are used for abruptly tapered fibers, whereas linear-index fiber modes provide the expansion basis for Kerr-type nonlinear-index fibers. Convergence of the results suggests in both cases that only a finite number of bound modes is sufficient to describe the field adequately. A comparison with simulations given by a finite-difference beam-propagation method that was developed for circularly symmetric waveguides confirms the validity of this assumption.

Abstract: Boussinesq equations describing one-dimensional unsteady, rapidly varied flows are integrated numerically to simulate both the sub- and supercritical flows and the formation of a hydraulic jump in a rectangular channel having a small bottom slope. For this purpose the MacCormack (second-order accurate in space and time) and two-four (second-order accurate in-time and fourth-order in space) explicit finite-difference schemes are used to solve the governing equations subject to specified end conditions until a steady state is reached. The inclusion of initial and boundary conditions is discussed, and the importance of the Boussinesq terms is investigated. Complete test results for a range of Froude numbers are presented that may be used by other researchers for the verification of mathematical models. A comparison of the computed measured results shows that the agreement between them is satisfactory for the fourth-order finite-difference scheme although the second-order scheme does not accurately predict the location of the jump. These simulations show that the Boussinesq terms have little effect in determining the location of the hydraulic jump.

Abstract: This paper describes a practical method in which irregular or locally irregular grids are used in reservoir simulation with the advantages of flexible approximation of reservoir geometry and reduced grid-orientation effects. Finite-difference equations are derived from an integral formulation of the reservoir model equations equivalent to the commonly used differential equations. Integrating over gridblocks results in material-balance equations for each block. This leads to a finite-volume method that combines the advantages of finite-element methods (flexible grids) with those of finite-difference methods (intuitive interpretation of flow terms). Grid-orientation effects are investigated. For grids based on triangular elements, the more isotropic distribution of gridpoints diminishes the orientation effect significantly. Numerical examples show that the regions of interest in a reservoir can be simulated efficiently and that well flow can be represented accurately.

Abstract: This study presents a mathematical formulation developed for aerodynamic sensitivity coefficients based on a discretized form of the compressible 2D Euler equations. A brief motivating introduction to the aerodynamic sensitivity analysis and the reasons behind an integrated flow/sensitivity analysis for design algorithms are presented. The finite difference approach and the quasi-analytical approach are used to determine the aerodynamic sensitivity coefficients. A new flow prediction concept, which is an outcome of the direct method in the quasi-analytical approach, is developed and illustrated with an example. Surface pressure coefficient distributions of a nozzle-afterbody configuration obtained from the predicted flowfield solution are compared successfully with their corresponding values obtained from a flowfield analysis code and the experimental data.

Abstract: Higher-order Pade approximations are applied to derive accurate and stable parabolic equations for sound propagation in oceans bounded below by an elastic bottom or bounded above by ice cover. Accuracy is achieved by placing constraints on the derivatives of the Pade approximations at the point corresponding to the reference wave number. Stability is achieved by requiring that the Pade approximations map part of the lower-left quadrant of the complex plane into the upper half of the complex plane. Elastic parabolic equations based on these Pade series can handle problems involving compressional, shear, and interface waves, very wide propagation angles, and large depth variations and weak range variations in the seismoacoustic parameters. A finite-difference spectral solution is developed for generating reference solutions and starting fields. The rotated elastic parabolic equation is used to investigate the accuracy of the elastic parabolic equation for range-dependent problems.

Abstract: A finite difference technique on rectangular cell-centered grids with local refinement is proposed in order to derive discretizations of second-order elliptic equations of divergence type approximating the so-called balance equa1 1/2 tion. Error estimates in a discrete H -norm are derived of order h ' for a simple symmetric scheme, and of order h ' for both a nonsymmetric and a more accurate symmetric one, provided that the solution belongs to H +a for a > \\ and a > \\ , respectively.

Abstract: Three different finite difference schemes for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval are considered. The schemes are based on the forward Euler, the backward Euler and the Crank-Nicolson methods. Error estimates are derived in maximum norm. Results from a numerical experiment are presented.

Abstract: A finite-difference time-domain (FD-TD) formulation is described. It is shown that the finite-difference time-domain formulation is equivalent to the symmetrical condensed node model used in the transmission line matrix (TLM) method. The TLM method can be formulated exactly in a finite-difference form in terms of total field quantities. It is shown that, due to a better field resolution and fulfilment of continuity conditions, the FD-TD formulation or its TLM equivalent model give better convergence and accuracy than the traditional FD-TD method. This is illustrated by numerical results pertaining to a finned waveguide. >

Abstract: A split-explicit finite difference scheme is developed which combines the accuracy and economy required for numerical weather prediction with the conservation properties required for climate-change experiments. Results are presented to demonstrate the scheme working in practice.

Abstract: An efficient and simple explicit finite difference beam propagation method (EFD-BPM) incorporating nonuniform mesh is described. The criteria for stability are developed, and it is shown that this algorithm is power conserving when the stability criteria are met. EFD-BPM is applied to the analysis of single and coupled semiconductor rib waveguides and its accuracy is confirmed by comparing the results with the reported results. Nonuniform mesh is found to improve the efficiency of the method significantly for the analysis of weakly guiding waveguide structures. Several coupled rib waveguide structures with curved input and output branching sections are analyzed using both three-dimensional EFD-BPM and two-dimensional finite difference BPM combined with effective index approximation. >

Abstract: High-order compact finite difference schemes for two-dimensional convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h4) schemes conform to a 3 × 3 stencil. We show that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h2) centred differencing and commonly used O(h) and O(h3) upwinding schemes.

Abstract: This paper presents analytical and numerical thermal analysis for melting and consolidating impregnated composite tapes in the presence of a localized heat source. This analysis also leads to the prediction of the processing window for a given tape-laying configuration. Heat of melting/solidification is included in the form of a heat generation term. A separation of variables method is employed to solve the governing equations ana lytically. In the numerical analysis, the governing equations are discretized using a non uniform mesh and are solved using a finite difference approach. The processing parame ters, such as consolidation speed, heat intensity, heat source width, etc., as well as material properties are incorporated within the analysis. The results show large thermal gradients in the vicinity of the consolidation point. The error between the analytical solu tion and the numerical result is found to be 3 % for the maximum temperature, and the maximum error for the temperature over the entire domai...

Abstract: The linearized unsteady aerodynamic response of a cascade of airfoils subjected to entropic, vortical, and acoustic gusts is analyzed. Field equations for the first-order unsteady perturbation flow are obtained by linearizing the full time-dependent mass, momentum, and energy conservation equations about a nonlinear, isentropic, and irrotational mean or steady flow. A splitting technique is then used to decompose the unsteady velocity field into irrotational and rotational parts leading to field equations for the unsteady entropy, rotational velocity, and irrotational velocity fluctuations that are coupled only sequentially. The entropic and rotational velocity fluctuations can be described in terms of the mean-flow drift and stream functions which can be computed numerically. The irrotational unsteady velocity is described by an inhomogeneous linearized potential equation which contains a source term that depends on the rotational velocity field. This equation is solved via a finite difference technique. Results are presented to indicate the status of the numerical solution procedure and to demonstrate the impact of blade geometry and mean blade loading on the aerodynamic response of cascades to vortical gust excitations. The analysis described leads to very efficient predictions of cascade unsteady aerodynamics phenomena making it useful for turbomachinery aeroelastic and aeroacoustic design applications.

Abstract: Part 1 Introduction to partial differential equations: behaviour physical systems definitions and equation properties conclusion problems. Part 2 Finite difference methods: discrete approximations in one dimension a generalized formulation of differences example calculations in one dimension solution of initial value problems finite differences multiple dimension two-dimensional example calculations conclusion problems. Part 3 Finite element methods: the methods of weighted residuals nomenclature computational procedures in one dimension mathematical requirements method of weighted residuals in two dimensions method of weighted residuals in three dimensions mathematical properties of the Galerkin finite element method conclusion. Part 4 Discretization considerations and design of approximations: spatial discretization temporal discretization space-time discretization alternative numerical procedures conclusion problems. Part 5 Accuracy and error reduction: improved accuracy through Mesh refinement improved accuracy through higher order approximation considerations of space-time problems conclusion problems.

Abstract: A global model which integrates three sequential steps for segmenting an image, namely, noise-filtering, local edge-detection, and integration of local edges into object boundaries, is described. The model overcomes some of the difficulties inherent in earlier global models, particularly their tendency to oversegment, and the lack of practical numerical algorithms for implementing them. The model consists of two coupled elliptic functionals, one for smoothing out the noise, and the other for boundary detection. The latter is obtained by regularizing the usual pointwise thresholding employed for boundary detection. The first variation of these functionals leads to coupled system of diffusion equations which are implemented by a simple finite difference scheme. The scheme may easily be converted into a parallel algorithm. >

Abstract: We present an a posteriori error estimator for the non-conforming Crouzeix-Raviart discretization of the Stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation. The error estimator yields global upper and local lower bounds for the error of the finite element solution. It can easily be generalized to the stationary, incompressible Navier-Stokes equations and to other non-conforming finite element methods. Numerical examples show the efficiency of the proposed error estimator.

Abstract: A study has been performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. Several existing sensitivity calculation methods and two new methods are compared for three example problems. All of the methods considered are computationally efficient enough to be suitable for largeorder finite element models. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the convergence of both response quantities and sensitivities as a function of the number of vectors used. Two types of sensitivity calculation techniques were considered. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. This was found to result in poor convergence of stress sensitivities in several cases. To overcome this poor convergence, two semianalytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in very good convergence of the stress sensitivities. In both techniques the computational cost is much less than would result if the vibration modes were recalculated and then used in an overall finite difference method. A dot over a symbol indicates derivative with respect to time. A superscriptT indicates a transposed matrix.