Showing papers on "Finite difference method published in 1990"
TL;DR: In this paper, the traditional finite difference time domain (FDTD) formulation is extended to include a discrete time-domain convolution, which is efficiently evaluated using recursion, and the accuracy of the extension is demonstrated by computing the reflection coefficient at an air-water interface over a wide frequency band including the effects of the frequency-dependent permittivity of water.
Abstract: The traditional finite-difference time-domain (FDTD) formulation is extended to include a discrete time-domain convolution, which is efficiently evaluated using recursion. The accuracy of the extension is demonstrated by computing the reflection coefficient at an air-water interface over a wide frequency band including the effects of the frequency-dependent permittivity of water. Extension to frequency-dependent permeability and to three dimensions is straightforward. The frequency dependent FDTD formulation allows computation of electromagnetic interaction with virtually any material and geometry (subject only to computer resource limitations) with pulse excitation. Materials that are highly dispersive, such as snow, ice, plasma, and radar-absorbing material, can be considered efficiently by using this formulation. >
687 citations
TL;DR: In this paper, a selfconsistent, one-dimensional solution of the Schrodinger and Poisson equations is obtained using the finite-difference method with a nonuniform mesh size.
Abstract: A self‐consistent, one‐dimensional solution of the Schrodinger and Poisson equations is obtained using the finite‐difference method with a nonuniform mesh size. The use of the proper matrix transformation allows preservation of the symmetry of the discretized Schrodinger equation, even with the use of a nonuniform mesh size, therefore reducing the computation time. This method is very efficient in finding eigenstates extending over relatively large spatial areas without loss of accuracy. For confirmation of the accuracy of this method, a comparison is made with the exactly calculated eigenstates of GaAs/AlGaAs rectangular wells. An example of the solution of the conduction band and the electron density distribution of a single‐heterostructure GaAs/AlGaAs is also presented.
674 citations
TL;DR: It is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an Eulerian-Lagrangian approach with large time steps is used to discretize the convective terms.
411 citations
TL;DR: In this article, a spatial discretization method for polar and nonpolar parabolic equations in one space variable is proposed, which is suitable for use in a library program.
Abstract: This paper is concerned with the design of a spatial discretization method for polar and nonpolar parabolic equations in one space variable. A new spatial discretization method suitable for use in a library program is derived. The relationship to other methods is explored. Truncation error analysis and numerical examples are used to illustrate the accuracy of the new algorithm and to compare it with other recent codes.
368 citations
TL;DR: In this article, the authors develop a rapid implicit solution technique for the enthalpy formulation of conduction controlled phase change problems, which is based on three existing implicit enthpy schemes.
Abstract: This paper develops a rapid implicit solution technique for the enthalpy formulation of conduction controlled phase change problems. Initially, three existing implicit enthalpy schemes are introduc...
337 citations
Book•
01 Apr 1990
TL;DR: This book presents the simplest description of continuous fluid flow, transport as a combination of addiction and diffusion, and solution methods for unsteady free surface flows.
Abstract: 1. Introduction. 2. The simplest description of continuous fluid flow. 3. The finite difference method. 4. Diffusion problems. 5. Transport as a combination of addiction and diffusion. 6. Descriptions of unsteady flows. 7. Solution methods for unsteady free surface flows. 8. Equilibrium methods. 9. Computational fluid dynamics of turbulence. 10. An introduction to some other numerical methods. Index.
311 citations
TL;DR: Results of numerical simulation are described to demonstrate the method and general features of the neural algorithms are discussed, including those for solving finite difference equations.
301 citations
TL;DR: MacCormack and Gabutti as mentioned in this paper introduced explicit finite-difference schemes to integrate the equations describing two-dimensional, unsteady gradually varied flows, which allow sharp discontinuous initial conditions, and do not require isolation of the bores.
Abstract: MacCormack and Gabutti explicit finite‐difference schemes are introduced to integrate the equations describing two‐dimensional, unsteady gradually varied flows. Both schemes are second‐order accurate in space and time, allow sharp discontinuous initial conditions, and do not require isolation of the bores. Both sub‐ and supercritical flows may be present simultaneously in different parts of the channel or in a sequence in time. The inclusion of boundaries and stability conditions and the addition of artificial viscosity to smooth high‐frequency oscillations are discussed. To illustrate application of the schemes in hydraulic engineering, two typical problems are solved and the results of different schemes are compared.
269 citations
TL;DR: In this paper, two antennas are considered, a cylindrical monopole and a conical monopole, which are driven through an image plane from a coaxial transmission line and analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method.
Abstract: Two antennas are considered, a cylindrical monopole and a conical monopole. Both are driven through an image plane from a coaxial transmission line. Each of these antennas corresponds to a well-posed theoretical electromagnetic boundary value problem and a realizable experimental model. These antennas are analyzed by a straightforward application of the finite-difference-time-domain (FD-TD) method. The computed results for these antennas are shown to be in excellent agreement with accurate experimental measurements for both the time domain and the frequency domain. The graphical displays presented for the transient near-zone and far-zone radiation from these antennas provide physical insight into the radiation process. >
242 citations
TL;DR: The numerical implementation of a systematic method for the exact boundary controllability of the wave equation, concentrating on the particular case of Dirichlet controls, is discussed.
Abstract: In this paper we discuss the numerical implementation of a systematic method for the exact boundary controllability of the wave equation, concentrating on the particular case of Dirichlet controls. The numerical methods described here consist in a combination of: finite element approximations for the space discretization; explicit finite difference schemes for the time discretization; a preconditioned conjugate gradient algorithm for the solution of the discrete problems; a pre/post processing technique based on a biharmonic Tychonoff regularization. The efficiency of the computational methodology is illustrated by the results of numerical experiments.
205 citations
TL;DR: In this paper, the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the NLS equation and some consequences for numerical computations are demonstrated.
Abstract: It has recently been demonstrated that standard discretizations of the cubic nonlinear Schrodinger (NLS) equation may lead to spurious numerical behavior. In particular, the origins of numerically induced chaos and the loss of spatial symmetry are related to the homoclinic structure associated with the NLS equation. In this paper, an analytic description of the homoclinic structure via soliton type solutions is provided and some consequences for numerical computations are demonstrated. Differences between an integrable discretization and standard discretizations are highlighted.
TL;DR: In this paper, a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered, where the integral term is treated by means of a convolution quadrature suggested by Lubich.
Abstract: A difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered. The integral term is treated by means of a convolution quadrature suggested by Lubich. Some results from Lubich’s discretized fractional calculus play a crucial role in proving consistency. The verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature. A stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper.
TL;DR: In this paper, two efficient finite difference methods for solving the Richards' equation in one dimension are presented, and their use in a range of soils and conditions is investigated, and the methods add points to the space grid as an infiltration or redistribution front advances.
Abstract: Two efficient finite difference methods for solving Richards' equation in one dimension are presented, and their use in a range of soils and conditions is investigated. Large time steps are possible when the mass-conserving mixed form of Richards' equation is combined with an implicit iterative scheme, while a hyperbolic sine transform for the matric potential allows large spatial increments even in dry, inhomogeneous soil. Infiltration in a range of soils can be simulated in a few seconds on a personal computer with errors of only a few percent in the amount and distribution of soil water. One of the methods adds points to the space grid as an infiltration or redistribution front advances, thus gaining considerably in efficiency over the other fixed grid method for infiltration problems. In 17-s computing, this advancing front method simulated infiltration, redistribution, and drainage for 50 days in an inhomogeneous soil with nonuniform initial conditions. Only 16 space and 21 time steps were needed for the simulation, which included early ponding with the development and dissipation of a perched water table.
Book•
01 Jan 1990
TL;DR: The Difference Calculus First-Order Difference Equations Linear Difference Equation with Constant Coefficients Linear Partial Different Equations (LPDE) Nonlinear Difference Equational Problems.
Abstract: The Difference Calculus First-Order Difference Equations Linear Difference Equations Linear Difference Equations with Constant Coefficients Linear Partial Difference Equations Nonlinear Difference Equations Problems Appendix Notes and References Bibliography Index
TL;DR: An efficient local mesh refinement algorithm, subdividing a computational domain to resolve fine dimensions in a time-domain-finite-difference (TD-FD) space-time grid structure, is discussed in this paper.
Abstract: An efficient local mesh refinement algorithm, subdividing a computational domain to resolve fine dimensions in a time-domain-finite-difference (TD-FD) space-time grid structure, is discussed. At a discontinuous coarse-fine mesh interface, the boundary conditions for the tangential and normal field components are enforced for a smooth transition of highly nonuniform held quantities. >
TL;DR: In this article, a modified finite volume method for solving Maxwell's equations in the time domain is presented, which allows the use of general nonorthogonal mixed-polyhedral grids, is a direct generalisation of the canonical staggered-grid finite difference method.
Abstract: A modified finite volume method for solving Maxwell's equations in the time-domain is presented. This method, which allows the use of general nonorthogonal mixed-polyhedral grids, is a direct generalisation of the canonical staggered-grid finite difference method. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) this method allows more accurate modeling of non-rectangular structures. The traditional “stair-stepped” boundary approximations associated with the orthogonal grid based finite difference methods ate avoided. Numerical results demonstrating the accuracy of this new method are presented.
TL;DR: In this paper, the Navier-Stokes equations are solved by a finite difference method, and the interface is kept sharp by front tracking, while the difference between the large-amplitude stages of flows initiated by two and three-dimensional perturbations is discussed.
Abstract: The fully three‐dimensional deformation of an interface between two fluids resulting from a Rayleigh–Taylor instability is studied numerically in the limit of weak stratification. The Navier–Stokes equations are solved by a finite difference method, and the interface is kept sharp by front tracking. The difference between the large‐amplitude stages of flows initiated by two‐ and three‐dimensional perturbations is discussed.
TL;DR: For frequency‐space 3-D finite‐difference migration and modeling, the compensation operator is implemented using either the phase‐shift, or phase‐ shift‐plus‐interpolation method, depending on the extent of lateral velocity.
Abstract: One-pass three-dimensional (3-D) depth migration potentially offers more accurate imaging results than does conventional two-pass migration, in variable velocity media. Conventional one-pass 3-D migration, done with the method of finite-difference inline and crossline splitting, however, creates large errors in imaging complex structures due to paraxial wave-equation approximation of the one-way wave equation, inline-crossline splitting, and finite-difference grid dispersion. After analyzing the finite-difference errors in conventional 3-D poststack wave field extrapolation, the paper presents a method that compensates for the errors and yet still preserves the efficiency of the conventional finite-difference splitting method. For frequency-space 3-D finite-difference migration and modeling, the compensation operator is implemented using the phase-shift method, or phase-shift plus interpolation method, depending on the extent of lateral velocity variations. The compensation operator increases the accuracy of handling steep dips, suppresses the inline and crossline splitting error, and reduces finite-difference grid dispersions. Numerical calculations show that the quality of 3-D migration and 3-D modeling is improved significantly with the finite-difference error compensation method presented in this paper. 13 refs., 7 figs.
TL;DR: The numerical analysis of spectral methods when non-constant coefficients appear in the equation, either due to the original statement of the equations or to take into account the deformed geometry is presented.
Abstract: The numerical analysis of spectral methods when non-constant coefficients appear in the equation, either due to the original statement of the equations or to take into account the deformed geometry, is presented. Particular attention is devoted to the optimality of the discretization even for low values of the discretization parameter. The effect of some overintegration is also addressed, in order to possibly improve the accuracy of the discretization.
TL;DR: Two distinctly different approaches have been used to simulate the movement of bands through a chromatographic column: the first is based on the mass balance equation which can be integrated numerically over time and space to give the elution profile as discussed by the authors.
Abstract: Two distinctly different approaches have been used to simulate the movement of bands through a chromatographic column One example of the first approach is the Craig distribution model, which replaces the continuous column with a specific number of discrete equilibration processes Thus it introduces the concept of (theoretical) plates into chromatography, but is not able to explain satisfactorily their significance The second approach is based on the mass balance equation which can be integrated numerically over time and space to give the elution profile In this paper we discuss the physical meaning of the numerical integration process followed by the finite difference methods
TL;DR: A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach (LEZOOM) is used to solve the advection-dispersion equation as mentioned in this paper.
Abstract: A Lagrangian-Eulerian method with zoomable hidden fine-mesh approach (LEZOOM), that can be adapted with either finite element or finite difference methods, is used to solve the advection-dispersion equation. The approach is based on automatic adaptation of zooming a hidden fine mesh in regions where the sharp front is located. Application of LEZOOM to four bench mark problems indicates that it can handle the advection-dispersion/diffusion problems with mesh Peclet numbers ranged from 0 to {infinity} and with mesh Courant numbers well in excess of 1. Difficulties that can be resolved with LEZOOM include numerical dispersion, oscillations, the clipping of peaks, and the effect of grid orientation. Nonuniform grid as well as spatial temporally variable flow pose no problems with LEZOOM. Both initial and boundary value problems can be solved accurately with LEZOOM. It is shown that although the mixed Lagrangian-Eulerian (LE) approach (LEZOOM without zooming) also produces excessive numerical dispersion as the upstream finite element (UFE) method, the LE approach is superior to the UFE method.
TL;DR: The semianalytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied in this article.
Abstract: The semianalytical method of design sensitivity analysis that is widely used for calculating derivatives of static response with respect to design variables for structures modeled by finite elements is studied. It is shown that the method can have serious accuracy problems for shape design variables in structures modeled by beam, plate, truss, frame, and solid elements. Errors are shown to be associated with an incompatibility of the sensitivity field with the structure. An error index is developed to test the accuracy of the semianalytical method. It characterizes the difference in errors between a general finite-difference method and the semianalytical method. A method for improving the accuracy of the semianalytical method (when possible) is provided. Examples are presented to demonstrate the use of the error index.
TL;DR: In this paper, the authors evaluated and compared three moving-grid methods for 1D problems, viz, the finite element method of Miller and co-workers, the method published by Petzold, and a method based on ideas adopted from DorIi and Drury.
TL;DR: Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization are investigated and numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation are reported on.
Abstract: We consider conjugate gradient type methods for the solution of large linear systemsA x=b with complex coefficient matrices of the typeA=T+i?I whereT is Hermitian and ? a real scalar. Three different conjugate gradient type approaches with iterates defined by a minimal residual property, a Galerkin type condition, and an Euclidean error minimization, respectively, are investigated. In particular, we propose numerically stable implementations based on the ideas behind Paige and Saunder's SYMMLQ and MINRES for real symmetric matrices and derive error bounds for all three methods. It is shown how the special shift structure ofA can be preserved by using polynomial preconditioning, and results on the optimal choice of the polynomial preconditioner are given. Also, we report on some numerical experiments for matrices arising from finite difference approximations to the complex Helmholtz equation.
TL;DR: In this paper, a finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations, which can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions).
Abstract: A new finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations. The present method uses conventional quadrilateral elements for the interpolation of velocity components, so that it can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions). A Picard iteration scheme with either flow rate or elasticity increment is used to treat the non-Newtonian stresses as pseudo-body forces, and an efficient and consistent predictor-corrector scheme is adopted for both the particle-tracking and strain tensor calculations. The new method has been used to simulate entry flows of polymer melts in circular abrupt contractions using the K-BKZ integral constitutive model. Results are in very good agreement with existing numerical data. The important question of mesh refinement and convergence for integral models in complex flow at high flow rate has also been addressed, and satisfactory convergence and mesh-independent results are obtained. In addition, the present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature.
TL;DR: In this article, the convection-diffusion equa- tion method is used for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip-tic partial differential equations.
Abstract: We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional ellip- tic partial differential equations. A prototype is the convection-diffusion equa- tion. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, com- bined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.
TL;DR: In this paper, the authors report numerical results obtained with finite difference ENO schemes for the model problem of the linear convection equation with periodic boundary conditions, and they find that the numerical solution does not converge uniformly and that an improved discretization can result in larger errors.
Abstract: We report numerical results obtained with finite difference ENO schemes for the model problem of the linear convection equation with periodic boundary conditions. For the test function sin(x), the spatial and temporal errors decrease at the rate expected from the order of local truncation errors as the discretization is refined. If we take sin4(x) as our test function, however, we find that the numerical solution does not converge uniformly and that an improved discretization can result in larger errors. This difficulty is traced back to the linear stability characteristics of the individual stencils employed by the ENO algorithm. If we modify the algorithm to prevent the use of linearly unstable stencils, the proper rate of convergence is reestablished. The way toward recovering the correct order of accuracy of ENO schemes appears to involve a combination of fixed stencils in smooth regions and ENO stencils in regions of strong gradients —a concept that is developed in detail in a companion paper by Shu (this issue, 1990).
TL;DR: In this article, a finite difference method is proposed to remove the need for staggered grids in fluid dynamic computations, which can be applied to free convection in a square cavity, one-dimensional flow through an actuator disk and plane stagnation flow.
Abstract: A new finite difference method, which removes the need for staggered grids in fluid dynamic computations, is presented. Pressure checkerboarding is prevented through a differencing scheme that incorporates the influence of pressure on velocity gradients. The method is implemented in a SIMPLE-type algorithm, and applied to three test problems: one-dimensional flow through an actuator disk, plane stagnation flow, and free convection in a square cavity. Good agreement is obtained between the numerical solutions and the corresponding analytical or benchmark solutions
TL;DR: In this paper, a control-volume, finite-element technique for coupling coarse grids with local fine meshes is described, where the pressure equation is treated in a finite element manner, while the mobility terms are upstream weighted in the usual way.
Abstract: This paper describes a control-volume, finite-element technique for coupling coarse grids with local fine meshes. The pressure is treated in a finite-element manner, while the mobility terms are upstream weighted in the usual way. This requires identification of the cell volume and edges that are consistent with the linear finite-element discretization of the pressure. To ensure that the pressure equation yields an M matrix, various conditions are required for the type of triangulation allowed. Because the form of the equations is similar to the usual finite-difference discretization, standard techniques can be used to solve the Jacobian. The local mesh-refinement method is demonstrated on some thermal reservoir simulation problems, and computational results are presented. Significant savings in execution times are obtained while predictions similar to global fine-mesh runs are given.
TL;DR: In this article, a finite-difference time-domain (FDTD) method for calculating the radar cross section (RCS) of a perfectly conducting target is presented, and the maximum cell size, the minimum number of external cells, and a method to eliminate field storage in the shielded internal volume of perfect conductors to reduce the computer storage requirements of FDTD are discussed.
Abstract: Several improvements to the finite-difference time-domain (FDTD) method for calculating the radar cross section (RCS) of a perfectly conducting target are presented. Sinusoidal and pulsed FDTD excitations are compared to determine an efficient method of finding the frequency response of targets. The maximum cell size, the minimum number of external cells, and a method to eliminate field storage in the shielded internal volume of perfect conductors to reduce the computer storage requirements of FDTD are discussed. The magnetic-field DC offset induced by surface currents on perfectly conducting objects is observed, and its effects are removed by postprocessing to achieve convergence of the RCS calculations. RCS calculations using the FDTD method in two dimensions are presented for both square and circular infinite cylinders illuminated by both transverse electric and transverse magnetic polarized plane waves. The RCS of a metal cube in three dimensions is also presented. Good agreement between FDTD calculations and theoretical values was achieved for all cases, and parameters necessary to achieve this agreement are examined. >