Showing papers on "Finite difference method published in 1987"
1,575 citations
TL;DR: In this article, a simple method of constructing adaptive grids is presented and the benefits are demonstrated by calculations of Sod's shock tube problem (G. A. Sod, J. Comput. Phys. 27, 1 (1978)) and of a supernova explosion.
341 citations
TL;DR: In this article, a generalized numerical dispersion analysis for wave equation computations is developed, which can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band.
Abstract: Conventional finite-difference operators for numerical differentiation become progressively inaccurate at higher frequencies and therefore require very fine computational grids. This problem is avoided when the derivatives are computed by multiplication in the Fourier domain. However, because matrix transpositions are involved, efficient application of this method is restricted to computational environments where the complete data volume required by each computational step can be kept in random access memory. To circumvent these problems a generalized numerical dispersion analysis for wave equation computations is developed. Operators for spatial differentiation can then be designed by minimizing the corresponding peak relative error in group velocity within a spatial frequency band. For specified levels of maximum relative error in group velocity ranging from 0.03% to 3%, differentiators have been designed that have the largest possible bandwidth for a given operator length. The relation between operator length and the required number of grid points per shortest wavelength, for a required accuracy, provides a useful starting point for the design of cost-effective numerical schemes. To illustrate this, different alternatives for numerical simulation of the time evolution of acoustic waves in three-dimensional inhomogeneous media are investigated. It is demonstrated that algorithms can be implemented that require fewer arithmetic and I/O operations by orders of magnitude compared to conventional second-order finite-difference schemes to yield results with a specified minimum accuracy.
339 citations
TL;DR: In this paper, a control-volume-based finite difference procedure with appropriate averaging for the diffusion coefficients is used to solve the coupling between the solid and fluid regions, and the analysis is extended to study the optimum spacing between heat sources for a fixed heat input and a desired maximum temperature at the heat source.
Abstract: Conjugate heat transfer for two-dimensional, developing flow over an array of rectangular blocks, representing finite heat sources on parallel plates, is considered. Incompressible flow over multiple blocks is modeled using the fully elliptic form of the Navier-Stokes equations. A control-volume-based finite difference procedure with appropriate averaging for the diffusion coefficients is used to solve the coupling between the solid and fluid regions. The heat transfer characteristics resulting from recirculating zones around the blocks are presented. The analysis is extended to study the optimum spacing between heat sources for a fixed heat input and a desired maximum temperature at the heat source.
216 citations
TL;DR: The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy and is evaluated by comparing its results to analytic solutions in two and three dimensions.
Abstract: Although there are acceptable methods for calculating whole body electromagnetic absorption, no completely acceptable method for calculating the local specific absorption rate (SAR) at points within the body has been developed. Frequency domain methods, such as the method of moments (MoM) have achieved some success; however, MoM requires computer storage on the order of (3N) 2 and computation time on the order of (3N) 3 where N is the number of cells. The finite-difference time-domain (FDTD) method has been employed extensively in calculating the scattering of metallic objects, and recently is seeing some use in calculating the interaction of EM fields with complex, lossy dielectric bodies. Since the FDTD method has storage and time requirements proportional to N, it presents an attractive alternative to calculating SAR distribution in large bodies. This paper describes the FDTD method and evaluates it by comparing its results to analytic solutions in two and three dimensions. The utility of the FDTD method is demonstrated by a 3D scan of the human torso. The results obtained demonstrate that the FDTD method is capable of calculating internal SAR distribution with acceptable accuracy. With the availability of supercomputers, such as the CRAY II, the calculation of SAR distribution in a man model of 50 000 cells (1.27 cm per cell) appears to be feasible.
183 citations
TL;DR: In this paper, a semi-implicit algorithm for the solution of the nonlinear, three-dimensional, resistive MHD equations in cylindrical geometry is presented, which assumes uniform density and pressure, although this is not a restriction of the method.
169 citations
TL;DR: In this paper, a linear model for neutral surface-layer flow over complex terrain is presented, which combines the simplicity and computational efficiency of linear methods with flexibility for closure schemes of finite-difference methods.
Abstract: A linear model for neutral surface-layer flow over complex terrain is presented. The spectral approach in the two horizontal coordinates and the finite-difference method in the vertical combines the simplicity and computational efficiency of linear methods with flexibility for closure schemes of finite-difference methods. This model makes it possible to make high-resolution computations for an arbitrary distribution of surface roughness and topography. Mixing-length closure as well as E − e closure are applied to two-dimensional flow above sinusoidal variations in surface roughness, the step-in-roughness problem, and to two-dimensional flow over simple sinusoidal topography. The main difference between the two closure schemes is found in the shear-stress results. E − e has a more realistic description of the memory effects in length and velocity scales when the surface conditions change. Comparison between three-dimensional model calculations and field data from Askervein hill shows that in the outer layer, the advection effects in the shear stress itself are also important. In this layer, an extra equation for the shear stress is needed.
160 citations
TL;DR: Finite-difference versions of some recently developed Krylov subspace projection methods are presented and analyzed in the context of solving systems of nonlinear equations using Inexact-Newton Met... as discussed by the authors.
Abstract: Finite-difference versions of some recently developed Krylov subspace projection methods are presented and analysed in the context of solving systems of nonlinear equations using Inexact-Newton Met...
134 citations
TL;DR: It appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods).
131 citations
TL;DR: In this paper, a program package was developed to evaluate electromagnetic fields inside arbitrary transmission-line connecting structures and to compute the scattering matrix, and detailed results were given and discussed regarding the fundamental behavior of embedding.
Abstract: The embedding of microwave devices is treated by applying the finite-difference method to three-dimensional shielded structures. A program package was developed to evaluate electromagnetic fields inside arbitrary transmission-line connecting structures and to compute the scattering matrix. The air bridge, the transition through a wall, and the bond wire are examined as interconnecting structures. Detailed results are given and discussed regarding the fundamental behavior of embedding.
125 citations
TL;DR: In this article, a modified equation analysis was used to develop formally fourth order accurate finite difference and pseudospectral methods for the one-dimensional wave equation, which can be used to achieve fourth order time accuracy with no increase in storage.
Abstract: In this paper we use a modified equation analysis to develop formally fourth order accurate finite difference and pseudospectral methods for the one-dimensional wave equation. The difference scheme is constructed by performing a modified equation analysis of a centered, second-order conservative scheme to determine its dominant error term. Subtracting a centered discretization of this term from the scheme cancels the second order truncation errors. This technique yields a formally fourth order accurate explicit difference scheme that employs only three time levels. Similarly, the modified equation technique can be used to achieve fourth order time accuracy for the pseudospectral method with no increase in storage. The difference and pseudospectral schemes are fourth order convergent for constant coefficients even when a spatially singular forcing term is used for a source. Numerical results are given comparing the accuracy and efficiency of these methods for some model problems. Finally, we present a gene...
Book•
01 Jun 1987
TL;DR: In this paper, the FEBS method is used to study the stability of boundary-value problems for Equations of Second Order and First and Second Order Difference Equations, respectively.
Abstract: Ordinary Difference Equations. Difference Equations of First and Second Order. Examples of Difference Schemes. Boundary-Value Problems for Equations of Second Order. Basis of the FEBS Method. Difference Schemes for Ordinary Differential Equations. Elementary Examples of Difference Schemes. Convergence of the Solutions of Difference Equations as a Consequence of Approximation and Stability. Widely-Used Difference Schemes. Difference Schemes for Partial Differential Equations. Basic Concepts. Simplest Examples of the Construction and Study of Difference Schemes. Some Basic Methods for the Study of Stability. Difference Scheme Concepts in the Computation of Generalized Solutions. Problems with Two Space Variable. The Concept of Difference Schemes with Splitting. Elliptic Problems. Concept of Variational-Difference and Projection-Difference Schemes. Stability of Evolutional Boundary-Value Problems Viewed as the Boundedness of Norms of Powers of a Certain Operator. Construction of the Transition Operator. Spectral Criterion for the Stability of Nonselfadjoint Evolutional Boundary-Value Problems. Appendix: Method of Internal Boundary Conditions. Bibliographical Commentaries. Bibliography. Index.
TL;DR: In this paper, Gabutti and Beam and Warming (BW) finite difference schemes are used for the analysis of free-surface flows resulting from the breaking of a dam.
Abstract: Two new finite-difference schemes - Gabutti, and Beam and Warming - are introduced and compared for the analysis of unsteady free-surface flows resulting from the breaking of a dam. These schemes split the fluxvector into positive and negative parts, each of which corresponds to the direction of a characteristic, thereby allowing use of proper finite differences for the space derivatives. Central finite differences are used for subcritical flow and upwind differences are used for supercritical flow. The details of these schemes are presented and the computed results are compared with the analytical solution to demonstrate their validity. Because of their simplicity, these schemes are attractive for solving the dam-break problem, especially when supercritical flow is present.
TL;DR: In this article, an analytical formulation for the computation of scattering and transmission by general anisotropic stratified material is presented, which employs a first-order state-vector differential equation representation of the Maxwell's equations whose solution is given in terms of a 4 \times 4 transition matrix relating the tangential field components at the input and output planes of the region.
Abstract: An analytical formulation is presented for the computation of scattering and transmission by general anisotropic stratified material. This method employs a first-order state-vector differential equation representation of Maxwell's equations whose solution is given in terms of a 4 \times 4 transition matrix relating the tangential field components at the input and output planes of the anisotropic region. The complete diffraction problem is solved by combining impedance boundary conditions at these interfaces with the transition matrix relationship. A numerical algorithm is described which solves the state-vector equation using finite differences. The validation of the resultant computer program is discussed along with example calculations.
Book•
14 Sep 1987
TL;DR: In this paper, the Buckingham Pi Theorem Scaling was used to scale up the Buckingham pi scaling method for dimensionality analysis in the context of multidimensional analysis of continuous systems, and the Laplace Equation Hyperbolic Equations Index.
Abstract: DIMENSIONAL ANALYSIS AND SCALING: Dimensional Analysis The Buckingham Pi Theorem Scaling PERTURBATION METHODS: Regular Perturbation Singular Perturbation Boundary Layer Analysis Two Applications CALCULUS OF VARIATIONS: Variational Problems Necessary Conditions for Extrema The Simplest Problem Generalizations Hamiltonian Theory Isoperimetric Problems EQUATIONS OF APPLIED MATHEMATICS: Partial Differential Equations The Diffusion Equation Classical Techniques Integral Equations WAVE PHENOMENA IN CONTINUOUS SYSTEMS: Wave Propagation Mathematical Models of Continua The Wave Equation Gasdynamics Fluid Motions in R3 STABILITY AND BIFURCATION: Intuitive Ideas One Dimensional Problems Two Dimensional Problems Hydrodynamic Stability SIMILARITY METHODS Invariant Variational Problems Invariant Partial Differential Equations The General Similarity Method DIFFERENCE METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS: Finite Difference Methods The Diffusion Equation The Laplace Equation Hyperbolic Equations Index.
TL;DR: In this paper, the authors compared the FD-TD method with the fast Fourier transform conjugate gradient method (FFT-CGM) for solving the 2-D Iossy dielectric cylinder problem for both the TM and TE incident polarizations.
Abstract: The need for high-resolution distributive dosimetry demands a numerical method capable of handling finely discretized, arbtrarily inhomogeneous models of biological bodies At present, two of the most promising methods in terms of numerical efficiency are the fast-Fourier-transform conjugate gradient method (FFT-CGM) and the finite-difference time-domain (FD-TD) method In this paper, these two methods are compared with respect to their ability to solve the 2-D Iossy dielectric cylinder problem for both the TM and TE incident polarizations Substantial errors are found in the FFT-CGM solutions for the TE case The source of these errors is explained and a modified method is developed which, although inefficient, alleviates the problem and illuminates the difficulties encountered in applying the pulse-basis method of moments to biological problems In contrast, the FD-TD method is found to yield excellent solutions for both polarizations This, coupled with the numerical efficiency of the FD-TD method, suggests that it is superior to the FFT-CGM for biological problems
TL;DR: In this paper, a method for obtaining the consolidation behavior of a layered soil subjected to strip, circular, or rectangular surface loadings, or subjected to fluid withdrawal due to pumping is presented.
Abstract: A method is presented for obtaining the consolidation behaviour of a layered soil subjected to strip, circular, or rectangular surface loadings, or subjected to fluid withdrawal due to pumping. The solution method involves applying a Fourier or Hankel transform to the field quantities along with a Laplace transformation. The effect of the Fourier or Hankel transform is to reduce a two- or three-dimensional problem or one involving axial symmetry, to one involving only a single spatial dimension. In cases where the soil is horizontally layered, this has great advantages over conventional methods, such as finite element or finite difference methods, since very little computer storage and data preparation time is required. Solution of the time dependent problem is achieved by applying a Laplace transformation to the field variables, obtaining solutions in Laplace transform space, and then numerically inverting the transformed solutions to obtain the real time behaviour. This eliminates the need for ‘marching type’ schemes where a solution is found from one at a previous time. By direct inversion of the Laplace transform, a solution may be obtained directly at any given time.
TL;DR: In this article, the total variation diminishing (TVD) finite difference scheme was interpreted as a Lax-Wendroff scheme plus an upwind weighted artificial dissipation term, which can be added to existing MacCormack method codes.
Abstract: In this paper we show that the total variation diminishing (TVD) finite difference scheme which was analysed by Sweby [8] can be interpreted as a Lax—Wendroff scheme plus an upwind weighted artificial dissipation term. We then show that if we choose a particular flux limiter and remove the requirement for upwind weighting, we obtain an artificial dissipation term which is based on the theory of TVD schemes, which does not contain any problem dependent parameters and which can be added to existing MacCormack method codes. Finally, we conduct numerical experiments to examine the performance of this new method.
TL;DR: In this article, the authors used a conservative form of the Arakawa type for the convective terms in the Navier-Stokes equations and compared the results with earlier ones of Moin and Kim.
TL;DR: In this paper, a numerical technique suitable for solving axisymmetric, unsteady free-boundary problems in fluid mechanics is presented, based on a finite-difference solution of the equations of motion on a moving orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary shape at any time.
Abstract: A brief description of a numerical technique suitable for solving axisymmetric, unsteady free‐boundary problems in fluid mechanics is presented. The technique is based on a finite‐difference solution of the equations of motion on a moving orthogonal curvilinear coordinate system, which is constructed numerically and adjusted to fit the boundary shape at any time. The initial value problem is solved using a fully implicit first‐order backward time differencing scheme in order to insure numerical stability. As an example of application, the unsteady deformation of a bubble in a uniaxial extensional flow for Reynolds numbers is considered in the range of 0.1≤R≤100. The computation shows that the bubble extends indefinitely if the Weber number is larger than a critical value (W>Wc). Furthermore, it is shown that a bubble may not achieve a stable steady state even at subcritical values of Weber number if the initial shape is sufficiently different from the steady shape. Finally, potential‐flow solutions as an ...
TL;DR: In this paper, a method was developed for the solution of the pressure Poisson equation, with Neumann boundary conditions, on a non-staggered grid, using primitive variables.
TL;DR: In this article, the construction of a spline function for a class of singular two-point boundary value problems is discussed, and three point finite difference methods are obtained for the solution of the boundary value problem.
Abstract: In this paper we discuss the construction of a spline function for a class of singular two-point boundary value problemx−α(xαu′)=f (x, u),u(0)=A,u(1)=B, 0<α<1 or α=1,2. The boundary conditions may also be of the formu′(0)=0,u(1)=B. Three point finite difference methods, using the above splines, are obtained for the solution of the boundary value problem. These methods are of second order and are illustrated by four numerical examples.
TL;DR: In this paper, Aanonsen et al. used a Fourier series expansion of the sound pressure to reduce the KZK equation to a set of coupled parabolic equations.
Abstract: Focused finite amplitude sound fields are investigated with numerical solutions of the Khokhlov‐Zabolotskaya‐Kuznetsov (KZK) equation. The numerical solution is based on the algorithm developed by Aanonsen et al. [J. Acoust. Soc. Am. 75, 749–768 (1984)], who used a Fourier series expansion of the sound pressure to reduce the KZK equation to a set of coupled parabolic equations. The basic algorithm has been modified by introducing a coordinate system that follows the convergent geometry of focused sound fields. In this way, more efficient numerical evaluation of the detailed field structure within the focal region is achieved. Arbitrary axisymmetric sources can be modeled. Here, circular sources having linear focusing gains of order 50 will be considered. The calculated time waveforms, propagation curves, and beam patterns illustrate clearly the combined effects of nonlinearity, diffraction, and absorption on finite amplitude sound that passes through a focal region. Among the new results are power curves ...
TL;DR: In this paper, the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method, where a motion of the shaft round the centered position is assumed.
Abstract: For modelling the turbulent flow in a seal the Navier-Stokes equations in connection with a turbulence (kappa-epsilon) model are solved by a finite-difference method. A motion of the shaft round the centered position is assumed. After calculating the corresponding flow field and the pressure distribution, the rotor-dynamic coefficients of the seal can be determined. These coefficients are compared with results obtained by using the bulk flow theory of Childs and with experimental results.
TL;DR: In this paper, the authors truncated the tensorial expansions using the serendipity approximation in an attempt to reduce the total number of unknowns and improve the effectiveness of the nodal collocation method.
TL;DR: In this article, it was shown that if the expanded node three-dimensional TLM method is operated in a certain way, then it can be numerically equivalent to a finite-difference method.
Abstract: It is shown that if the expanded node three-dimensional TLM method is operated in a certain way, then it can be numerically equivalent to a finite-difference method. Some comments are made on comparisons between the two approaches.
TL;DR: In this paper, the Navier-Stokes equations in axisymmetric cylindrical coordinates were used for predicting detailed flow patterns and temperature profiles during natural convection heating of canned liquids.
Abstract: Amathematical model was developed for the first time for predicting detailed flow patterns and temperature profiles during natural convection heating of canned liquids. Finite difference methods were used to solve the governing Navier-Stokes equations in axisymmetric cylindrical coordinates. A vorticity-stream function formulation of the equations was used. Details of the numerical techniques used are discussed. Plots of transient isotherms, streamlines and velocities are provided. From the standpoint of food processing, the slowest heating points migrated within the bottom 15% of the can with no particular pattern of migration.
TL;DR: In this paper, it was shown that an exact solution for offset dual-or single-shaped synthesis exists, and that an infinite set of such solutions exists, in part from numerical results.
Abstract: Since Kinber (Radio Technika and Engineering-1963) and Galindo (IEEE Trans. Antennas Propagat.-1963/1964) developed the solution to the circular symmetric dual shaped synthesis problem, the question of existence (and of uniqueness) for offset dual (or single) shaped synthesis has been a point of controversy. Many researchers thought that the exact offset solutions may not exist. Later, Galindo-Israel and Mittra (IEEE Trans. Antennas Propagat.-1979) and others formulated the problem exactly and obtained excellent and numerically efficient but approximate solutions. Using a technique similar to that first developed by Schruben for the single reflector problem (Journal of the Optical Society-1973), Brickell and Westcott (Proc. Institute of Electrical Engineering-1981) developed a Monge-Ampere (MA) second-order nonlinear partial differential equation for the dual reflector problem. They solved an elliptic form of this equation by a technique introduced by Rall (1979) which iterates, by a Newton method, a finite difference linearized MA equation. The elliptic character requires a set of finite difference equations to be developed and solved iteratively. Existence still remained in question. Although the second-order MA equation developed by Schruben is elliptic, the first-order equations from which the MA equation is derived can be integrated progressively (e.g., as for an initial condition problem such as for hyperbolic equations) a noniterative and usually more rapid type solution. In this paper, we have solved, numerically, the first-order equations. Exact solutions are thus obtained by progressive integration. Furthermore, we have concluded that not only does an exact solution exist, but an infinite set of such solutions exists. These conclusions are inferred, in part, from numerical results.