Showing papers on "Finite difference method published in 1997"
TL;DR: In this paper, a model for studying ocean circulation problems taking into account the complicated outline and bottom topography of the World Ocean is presented, and the model is designed to be as consistent as possible with the continuous equations with respect to energy.
1,048 citations
TL;DR: A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced, applicable to incompressible 2D flow simulations on finite volume and difference meshes and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner.
Abstract: SUMMARY A new algorithm for volume tracking which is based on the concept of flux-corrected transport (FCT) is introduced. It is applicable to incompressible 2D flow simulations on finite volume and difference meshes. The method requires no explicit interface reconstruction, is direction-split and can be extended to 3D and orthogonal curvilinear meshes in a straightforward manner. A comparison of the new scheme against well-known existing 2D finite volume techniques is undertaken. A series of progressively more difficult advection tests is used to test the accuracy of each scheme and it is seen that simple advection tests are inadequate indicators of the performance of volume-tracking methods. A straightforward methodology is presented that allows more rigorous estimates to be made of the error in volume advection and coupled volume and momentum advection in real flow situations. The volume advection schemes are put to a final test in the case of Rayleigh‐Taylor instability. 1997 by CSIRO. In the numerical computation of multifluid problems such as density currents or Rayleigh‐Taylor instability there is a need for an accurate representation of the interface separating two immiscible fluids. Free surface flows such as water waves and splashing droplets are an approximation to the multifluid problem in which one of the fluids (usually a gas) is neglected as having an insignificant influence on the dynamics of the system. In a general free surface flow problem, fluid coalescence and detachment may occur and deforming meshes cannot be used. In this case the need of an accurate and sharp interface is even greater than in true multifluid computations. Although a slightly diffuse interface may be acceptable in a problem where the continuity, momentum and energy equations are solved throughout the entire mesh, in a free surface simulation the location of the interface determines the size and shape of the computational domain and specifies where boundary conditions must be applied. In this case a diffuse interface cannot be tolerated. On finite volume (or difference) meshes, standard advection techniques can be used in multifluid problems to advect either the density or a material indicator function, however these methods are either diffusive (e.g. first order upwinding) or unstable (higher order schemes in which unphysical oscillations appear in the vicinity of the interface). Numerous techniques have been devised to limit the diffusiveness of low order schemes and to minimize the instability of high order schemes (see e.g.
782 citations
TL;DR: In this paper, it was shown that discrepancies between the results of dealiased spectral and standard nondialiased finite-difference methods are due to both aliasing and truncation errors with the latter being the leading source of differences.
564 citations
TL;DR: A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented and an implicit quasi-Newton method is developed that allows reasonable time steps to be used.
Abstract: A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019--1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.
495 citations
TL;DR: In this paper, a finite-difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two-dimensional Poisson equation, where  is a function of � or its derivatives,Sis a function for position, and  or its normal derivative is specified on the boundary.
470 citations
TL;DR: In this article, a conformal finite-difference time-domain (CFDTD) technique for locally distorted contours that accurately model curved metallic objects is presented, which yields results that are far more accurate than those generated by the conventional staircasing approach.
Abstract: A novel conformal finite-difference time-domain (CFDTD) technique for locally distorted contours that accurately model curved metallic objects is presented in this paper. This approach is easy to implement and is numerically stable. Several examples are presented to demonstrate that the new method yields results that are far more accurate than those generated by the conventional staircasing approach. Example geometries include cylindrical and spherical cavities, and a circular microstrip patch antenna. Accuracy of the scheme is demonstrated by comparing the results derived from analytical and Method of Moments (MoM) techniques.
404 citations
TL;DR: In this paper, the authors simulated flow past a circular cylinder at a Reynolds number of 3.9 X 10 3 using a solver that employs an energy-conservative second-order central difference scheme for spatial discretization.
Abstract: We have simulated flow past a circular cylinder at a Reynolds number of 3.9 X 10 3 using a solver that employs an energy-conservative second-order central difference scheme for spatial discretization. Detailed comparisons of turbulence statistics and energy spectra in the downstream wake region (7.0 < x/D < 10.0) have been made with the results of Beaudan and Moin and with experiments to assess the impact of numerical diffusion on the flowfield. Based on these comparisons, conclusions are drawn on the suitability of higher-order upwind schemes for LES in complex geometries.
397 citations
TL;DR: This paper considers the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems, and shows that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled.
Abstract: In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.
396 citations
TL;DR: An expanded mixed finite element approximation of second-order elliptic problems containing a tensor coefficient is presented, and it is shown that rates of convergence are retained for the finite difference method.
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.
381 citations
TL;DR: An hierarchy of uniformly high-order accurate schemes is presented which generalizes Godunov's scheme and its second- order accurate MUSCL extension to an arbitrary order of accuracy.
353 citations
TL;DR: In this article, a flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented, where the 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme.
Abstract: A flexible and simple way of introducing stress-free boundary conditions for including three-dimensional (3D) topography in the finite-difference method is presented. The 3D topography is discretized in a staircase by stacking unit material cells in a staggered-grid scheme. The shear stresses are distributed on the 12 edges of the unit material cell so that only shear stresses appear on the free surface and normal stresses always remain embedded within the solid region. This configuration makes it possible to implement stress-free boundary conditions at the free surface by setting the Lame coefficients λ and μ to zero without generating any physically unjustified condition. Arbitrary 3D topographies are realized by changing the distribution of λ and μ in the computational domain. Our method uses a parsimonious staggered-grid scheme that requires only 3/4 of the memory used in the conventional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finite-difference results are compared with those of the boundary-element method for the two-dimensional (2D) semi-circular canyon model. We also present the responses of a segment of semi-circular canyon and hemispherical cavity to vertically incident plane P, SV , and SH waves and discuss the response of a Gaussian hill to an isotropic point source embedded in the hill. In the segment of semi-circular canyon, the later portions of the synthetics are characterized by phases scattered from the two vertical side walls. The hemispherical cavity and 2D semi-circular canyon both show focusing of energy at the bottom of the cavity, although the focusing effect is stronger in the former geometry. Focusing and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot located on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.
Book•
18 Sep 1997
TL;DR: In this article, the inverse problem for difference equations and the ring S of sequences of sequences are considered. But they do not address the problem of difference modules over the canonical forms of difference equations.
Abstract: Picard-Vessiot rings.- Algorithms for difference equations.- The inverse problem for difference equations.- The ring S of sequences.- An excursion in positive characteristic.- Difference modules over .- Classification and canonical forms.- Semi-regular difference equations.- Mild difference equations.- Examples of equations and galois groups.- Wild difference equations.- q-difference equations.
TL;DR: In this paper, a new approach based on a new property of phase-lag for computing eigenvalues of Schrodinger equations with potentials, is developed in two cases: (i) the specific case in which the potential V(x) is an even function with respect to x.
TL;DR: In this article, a general formulation for finite-difference time-domain (FDTD) modeling of wave propagation in arbitrary frequency-dispersive media is presented, and two algorithmic approaches are outlined for incorporating dispersion into the FDTD time-stepping equations.
Abstract: A general formulation is presented for finite-difference time-domain (FDTD) modeling of wave propagation in arbitrary frequency-dispersive media. Two algorithmic approaches are outlined for incorporating dispersion into the FDTD time-stepping equations. The first employs a frequency-dependent complex permittivity (denoted Form-1), and the second employs a frequency-dependent complex conductivity (denoted Form-2). A Pade representation is used in Z-transform space to represent the frequency-dependent permittivity (Form-1) or conductivity (Form-2). This is a generalization over several previous methods employing either Debye, Lorentz, or Drude models. The coefficients of the Pade model may be obtained through an optimization process, leading directly to a finite-difference representation of the dispersion relation, without introducing discretization error. Stability criteria for the dispersive FDTD algorithms are given. We show that several previously developed dispersive FDTD algorithms can be cast as special cases of our more general framework. Simulation results are presented for a one-dimensional (1-D) air/muscle example considered previously in the literature and a three-dimensional (3-D) radiation problem in dispersive, lossy soil using measured soil data.
Journal Article•
TL;DR: In this paper, a method based on an interpolating wavelet transform using polynomial interpolation on dyadic grids is presented for adaptively solving hyperbolic PDEs.
Abstract: A method is presented for adaptively solving hyperbolic PDEs. The method is based on an interpolating wavelet transform using polynomial interpolation on dyadic grids. The adaptability is performed automatically by thresholding the wavelet coefficients. Operations such as differentiation and multiplication are fast and simple due to the one-to-one correspondence between point values and wavelet coefficients in the interpolating basis. Treatment of boundary conditions is simplified in this sparse point representation (SPR). Numerical examples are presented for one- and two-dimensional problems. It is found that the proposed method outperforms a finite difference method on a uniform grid for certain problems in terms of flops.
TL;DR: In this paper, an implementation of analytic second derivatives with respect to nuclear displacements, based on density functional theory within the Kohn-Sham formalism, is reported. But the efficiency of the algorithm is tested in comparison with finite difference methods.
TL;DR: In this article, the authors gave an account of some new results on conformal FDTD obtained by the authors and their associates at Lockheed Martin Space Company since 1988, focusing on nonsmooth boundary condition simulation.
Abstract: The finite-difference time-domain (FDTD) and its current generalizations have been demonstrated to be useful and powerful tools for the calculation of the radar cross section (RCS) of complicated objects, the radiation of antennas in the presence of other structures, and other applications. The mathematical techniques for conformal FDTD have matured; the primary impediments to its implementation are the complex geometries and material properties associated with the problem. Even under these circumstances, FDTD is more flexible to implement because it is based on first principles instead of a clever mathematical trick. This paper gives an account of some new results on conformal FDTD obtained by the authors and their associates at Lockheed Martin Space Company since 1988. The emphasis is on nonsmooth boundary condition simulation.
TL;DR: In this paper, an algorithm for the numerical modeling of magnetotelluric fields in 2D generally anisotropic block structures is presented, where electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor.
Abstract: SUMMARY An algorithm for the numerical modelling of magnetotelluric fields in 2-D generally anisotropic block structures is presented. Electrical properties of the individual homogeneous blocks are described by an arbitrary symmetric and positive-definite conductivity tensor. The problem leads to a coupled system of partial differential equations for the strike-parallel components of the electromagnetic field, Ex and H,. These equations are numerically approximated by the finite-difference (FD) method, making use of the integro-interpolation approach. As the magnetic component H, is constant in the non-conductive air, only equations for the electric mode are approximated within the air layer. The system of linear difference equations, resulting from the FD approximation, can be arranged in such a way that its matrix is symmetric and band-limited, and can be solved, for not too large models, by Gaussian elimination. The algorithm is applied to model situations which demonstrate some non-trivial phenomena caused by electrical anisotropy. In particular, the effect of 2-D anisotropy on the relation between magnetotelluric impedances and induction arrows is studied in detail.
TL;DR: In this article, a compact central-difference approximation in conjunction with the Yee (1966) grid is used to compute the spatial derivatives in Maxwell's equations, and the four-stage Runge-Kutta (RK) integrator is invoked.
Abstract: A compact central-difference approximation in conjunction with the Yee (1966) grid is used to compute the spatial derivatives in Maxwell's equations. To advance the semi-discrete equations, the four-stage Runge-Kutta (RK) integrator is invoked. This combination of spatial and temporal differencing leads to a scheme that is fourth-order accurate, conditionally stable, and highly efficient. Moreover, the use of compact differencing allows one to apply the compact operator in the vicinity of a perfect conductor-an attribute not found in other higher order methods. Results are provided that quantify the spectral properties of the method. Simulations are conducted on problem spaces that span one and three dimensions and whose domains are of the open and closed type. Results from these simulations are compared with exact closed-form solutions; the agreement between these results is consistent with numerical analysis.
TL;DR: The ideas of algebraic topology are applied to the analysis of the finite volume and finite element methods, illuminating the similarity between the discretization strategies adopted by the two methods, in the light of a geometric interpretation proposed for the role played by the weighting functions in finite elements.
TL;DR: Simulation results with two-dimensional test media have shown that this method can yield quantitatively as well as qualitatively accurate reconstructions of absorption and scattering distributions in cases in which the first-order Born approximation cannot work well.
Abstract: We present a Born iterative method for reconstructing optical properties of turbid media by means of frequency-domain data. The approach is based on iterative solutions of a linear perturbation equation, which is derived from the integral form of the Helmholtz wave equation for photon-density waves. In each iteration the total field and the associated weight matrix are recalculated based on the previous reconstructed image. We then obtain a new estimate by solving the updated perturbation equation. The forward solution, also based on a Helmholtz equation, is obtained by a multigrid finite difference method. The inversion is carried out through a Tikhonov regularized optimization process by the conjugate gradient descent method. Using this method, we first reconstruct the distribution of the complex wave numbers in a test medium, from which the absorption and the scattering distributions are then derived. Simulation results with two-dimensional test media have shown that this method can yield quantitatively (in terms of coefficient values) as well as qualitatively (in terms of object location and shape) accurate reconstructions of absorption and scattering distributions in cases in which the first-order Born approximation cannot work well. Both full-angle and limited-angle measurement schemes have been simulated to examine the effect of the location of detectors and sources. The robustness of the algorithm to noise has also been evaluated.
TL;DR: A mathematical model for coupled heat and moisture transfer problem is presented in this article, which consists of four partial differential equations for mass balance, heat balance and drying rate of grain drying.
Abstract: Grain drying is a simultaneous heat and moisture transfer problem. The modelling of such a problem is of significance in understanding and controlling the drying process. In the present study, a mathematical model for coupled heat and moisture transfer problem is presented. The model consists of four partial differential equations for mass balance, heat balance, heat transfer and drying rate. A simple finite difference method is used to solve the equations. The method shows good flexibility in choosing time and space steps which enable the simulation of long term grain drying/cooling processes. A deep barley bed is used as an example of grain beds in the current simulation. The results are verified against experimental data taken from literature. The analysis of the effects of operating conditions on the temperature and moisture content within the bed is also carried out
TL;DR: In this paper, an adaptive finite difference scheme for the solution of the discrete first order Hamilton-Jacobi-Bellman equation is presented, based on local a posteriori error estimates and certain properties of these estimates are proved.
Abstract: In this paper an adaptive finite difference scheme for the solution of the discrete first order Hamilton-Jacobi-Bellman equation is presented. Local a posteriori error estimates are established and certain properties of these estimates are proved. Based on these estimates an adapting iteration for the discretization of the state space is developed. An implementation of the scheme for two-dimensional grids is given and numerical examples are discussed.
TL;DR: In this paper, volume-averaged equations are developed governing steady, laminar, fully developed, hydromagnetic mixed convection non-Darcian flow of an electrically conducting and heat-generating / absorbing fluid in a channel embedded in a uniform porous medium.
Abstract: Volume-averaged equations are developed governing steady, laminar, fully developed, hydromagnetic mixed convection non-Darcian flow of an electrically conducting and heat-generating / absorbing fluid in a channel embedded in a uniform porous medium. Proper dimensionless parameters are employed for various thermal boundary conditions on the left and right walk of the channel prescribed as isothermal-isothermal, isothermal-iso-flux, and isoflux-isothermal. Analytical expressions for the velocity and temperature profiles in the channel as well as for the mass flow rate, friction factor, and heat carried out by the fluid in the channel are developed for special cases of the problem. Conditions for the occurrence of fluid backflow zones are reported. The fully nonlinear governing equations are solved numerically by an implicit finite difference method. Favorable comparisons with the developed analytical results and previously published work are performed. Graphical results of the closed-form and numer...
TL;DR: In this paper, it is shown that the time-dependent sensitivity can be derived directly from the adjoint model solution, that is, the sensitivity to initial conditions as function of the initial time.
Abstract: Adjoint models are used for atmospheric and oceanic sensitivity studies in order to efficiently evaluate the sensitivity of a cost function (e.g., the temperature or pressure at some target time tf, averaged over some region of interest) with respect to the three-dimensional model initial conditions. The time-dependent sensitivity, that is the sensitivity to initial conditions as function of the initial time ti, may be obtained directly and most efficiently from the adjoint model solution. There are two approaches to formulating an adjoint of a given model. In the first (“finite difference of adjoint”), one derives the continuous adjoint equations from the linearized continuous forward model equations and then formulates the finite-difference implementation of the continuous adjoint equations. In the second (“adjoint of finite difference”), one derives the finite-difference adjoint equations directly from the finite difference of the forward model. It is shown here that the time-dependent sensiti...
TL;DR: In this article, the free convection flow of an electrically conducting fluid along a vertical plate embedded in a thermally stratified porous medium in the presence of a uniform normal magnetic field is investigated.
TL;DR: Using the finite difference method, a discretization scheme is presented and a convergence theorem for the first order derivatives is proved and some regularity properties of price functions are derived.
Abstract: We discuss pricing formulae for American options in Merton's jump-diffusion model. With the help of variational inequalities, we derive some regularity properties of price functions. Using the finite difference method, a discretization scheme is presented and a convergence theorem for the first order derivatives is proved. Numerical methods and results are also discussed.
TL;DR: The formulation combines the primitive function approach with five-point spatially sixth- and fourth-order methods to develop a fully discrete scheme for linear wave propagation phenomena with particular emphasis on computational electromagnetics in the time-domain.
TL;DR: In this article, variable space grid and boundary immobilization techniques based on the explicit finite difference are applied to the one-phase classical Stefan problem, and all the results obtained by the two methods are in good agreement with the exact solution, and exhibit the expected convergence as the mesh size is refined.
TL;DR: The framework needed to apply modern high accuracy numerical methods from computational gas dynamics to this extended system of convection-diffusion equations with stiff source terms is developed.