Showing papers on "Finite difference method published in 1984"
TL;DR: In this paper, a new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium, which uses both velocity and stress in a discrete grid.
Abstract: A new finite-difference (FD) method is presented for modeling SH-wave propagation in a generally heterogeneous medium. This method uses both velocity and stress in a discrete grid. Density and shear modulus are similarly discretized, avoiding any spatial smoothing. Therefore, boundaries will be correctly modeled under an implicit formulation. Standard problems (quarter-plane propagation, sedimentary basin propagation) are studied to compare this method with other methods. Finally a more complex example (a salt dome inside a two-layered medium) shows the effect of lateral propagation on seismograms recorded at the surface. A corner wave, always in-phase with the incident wave, and a head wave will appear, which will pose severe problems of interpretation with the usual vertical migration methods.
835 citations
TL;DR: In this article, the authors present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes, which consists of a combination of well-known numerical procedures such as Sturm sequences, the bisection method, Newton's and Brents methods, Richardson extrapolation, and inverse iteration.
Abstract: The method of normal modes is frequently used to solve acoustic propagation problems in stratified oceans. The propagation numbers for the modes are the eigenvalues of the boundary value problem to determine the depth dependent normal modes. Errors in the numerical determination of these eigenvalues appear as phase shifts in the range dependence of the acoustic field. Such errors can severely degrade the accuracy of the normal mode representation, particularly at long ranges. In this paper we present a fast finite difference method to accurately determine these propagation numbers and the corresponding normal modes. It consists of a combination of well‐known numerical procedures such as Sturm sequences, the bisection method, Newton’s and Brent’s methods, Richardson extrapolation, and inverse iteration. We also introduce a modified Richardson extrapolation procedure that substantially increases the speed and accuracy of the computation.
180 citations
TL;DR: In this article, a comprehensive study regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection-diffusion equation is presented.
Abstract: A comprehensive study is presented regarding the numerical stability of the simple and common forward Euler explicit integration technique combined with some common finite difference spatial discretizations applied to the advection-diffusion equation. One-dimensional results are obtained using both the matrix method (for several boundary conditions) and the classical von Neumann method of stability analysis and arguments presented showing that the latter is generally to be preferred, regardless of the type of boundary conditions. The less-well-known Godunov-Ryabenkii theory is also applied for a particular (Robin) boundary condition. After verifying portions of the one-dimensional theory with some numerical results, the stabilities of the two- and three-dimensional equations are addressed using the von Neumann method and results presented in the form of a new stability theorem. Extension of a useful scheme from one dimension, where the pure advection limit is known variously as Leith's method or a Lax-Wendroff method, to many dimensions via finite elements is also addressed and some stability results presented.
178 citations
TL;DR: In this paper, a numerical scheme is developed which automatically locates the angle at which a shock might be expected to cross the computing grid then constructs separate finite difference formulas for the flux components normal and tangential to this direction.
139 citations
TL;DR: In this paper, a 6th order method was developed for approximate numerical integration of the Schrodinger equation, which was shown to be generally superior to the widely used Numerov method.
134 citations
TL;DR: A method of mathematically analyzing interdendritic microsegregation was established using finite difference method and taking into consideration the diffusion of the solute in the solid and liquid phases as discussed by the authors.
Abstract: A method of mathematically analyzing interdendritic microsegregation was established using finite difference method and taking into consideration the diffusion of the solute in the solid and liquid phases. The cross-sectional shape of dendrites and the fact that the enrichment of the solute in the liquid phase at the solid-liquid interface restrains the advancement speed of the solid-liquid interface were considered. Directional solidification tests to examine interdendritic segregation were made to verify the mathematical analysis method established. The advantages of the new method over other methods were discussed. Then, spot-like segregations were mathematically analyzed applying the same method, and the results were in good agreement with the observations in continuously-cast slabs.
129 citations
TL;DR: In this paper, a method for computing the modes of dielectric guiding structures based on finite differences is described, and the numerical computation program is efficient and can be applied to a wide range of problems.
Abstract: A method for computing the modes of dielectric guiding structures based on finite differences is described. The numerical computation program is efficient and can be applied to a wide range of problems. We report here solutions for circular and rectangular dielectric waveguides and compare our solutions with those obtained by other methods. Limitations in the commonly used approximate formulas developed by Marcatili are discussed.
120 citations
TL;DR: In this article, the magnitude of the current at a plane electrode in a rectangular channel under conditions of steady state, fully developed laminar flow is numerically computed based on the backward implicit finite difference numerical method, applied to solve the differential equation governing convective diffusion.
119 citations
TL;DR: The boundary integral equation method (BIEM) for porous elasticity is proposed in this article, where the fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material.
Abstract: For physical phenomena governed by the Biot model of porous-elasticity, a reciprocal relation, similar to the Betti's recoprocal theorem in elasticity, is constructed in Laplace transformed space. Integrating the reciprocal relation enables one to formulate boundary integral equations. The fundamental kernels for the integral equations are solved in closed forms for the case of isotropic material. Numerical implementation of two-dimensional problems includes finite element ideas of discretization and polynomial interpolation, and numerical inversion of a Laplace transform. Practical applications of the method are found in consolidation problems in soils which contain compressible as well as incompressible pore fluids. Also, as a numerical experiment, consolidation of partially saturated soil is simulated and interesting phenomena are observed. The currently developed boundary integral equation method (BIEM) for porous-elasticity may be viewed as an efficient and accurate alternative of existing finite element and finite difference methods. For linear consolidation problems, application of BIEM is always preferred to the other numerical methods whenever possible.
118 citations
TL;DR: In this paper, a new method for computing the complete elastic response of a vertically heterogeneous half-space is presented, which utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series.
Abstract: Summary.
A new method is presented for computing the complete elastic response of a vertically heterogeneous half-space. The method utilizes a discrete wavenumber decomposition for the horizontal dependence of the wave motion in terms of a Fourier-Bessel series. The series representation is exact if summed to infinity and consequently eliminates the need to integrate a continuous Bessel transform numerically. In practice, a band-limited solution is obtained by truncating the series at large wavenumbers. The vertical and time dependence of the wave motion is obtained as the solution to a system of partial differential equations. These equations are solved numerically by a combination of finite element and finite difference methods which accommodate arbitrary vertical heterogeneities. By using a reciprocity relation, the wave motion is computed simultaneously for all source-observer combinations of interest so that the differential equations need only be solved once. A comparison is made, for layered media, between the solutions obtained by discrete wavenumber/finite element, wavenumber integration, axisymmetric finite element, and generalized rays.
116 citations
TL;DR: In this paper, a theory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary, where instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0.
Abstract: A th00 eory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity C ≥ 0. To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in l2 norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundstrom. In particular we investigate l2-instability with respect to both initial and boundary data and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with C = 0 versus C > 0.
TL;DR: A ‘modified equation’ analysis is used to characterize grid-dependent anisotropies in the discretization and motivates a new family of finite difference schemes that show essentially no dependence on the orientation of the grid.
TL;DR: In this paper, a numerical method to simulate discharging processes in mass flow silos is presented, which provides transient velocity and stress fields within the bulk material for a first period of discharging.
Abstract: A numerical method to simulate discharging processes in mass‐flow silos is presented. The essential point is to formulate the appropriate constitutive law for a granular bulk material, which covers solid‐like as well as fluid‐like behavior during discharging. An elastic‐plastic law is chosen for the former one, which is completed with a simple first approach for fluid‐like behavior. As large and fast deformations occur, geometric nonlinearities and mass properties of the bulk material are considered with respect to an Eulerian frame of reference. The complete set of field equations is numerically solved by the finite element method spatially and by the finite difference method in time. Due to the nature of the finite element method a broad variety of boundary conditions can be studied. The method provides transient velocity and stress fields within the bulk material for a first period of discharging. Remarkable stress redistributions with strong increases of wall pressures are computed.
TL;DR: In this article, a full implicit continuous Eulerian (FICE) scheme is developed for solving multidimensional transient MHD flow problems, where the boundary conditions are treated by classifying them into physical and computational ones.
TL;DR: In this paper, the second-order extension of the Godunov method was used for the solution of two-dimensional Euler equations in the subsonic, transonic, and supersonic flow regimes.
Abstract: The Godunov method and a new second-order accurate extension of the method are used for the solution of two-dimensional Euler equations. Both numerical schemes are described in detail. Their performances in the subsonic, transonic, and supersonic flow regimes are first tested on the problem of flow in a channel with a circular arc bump. The niethods are then applied to calculate the transonic flow through a supercritical com pressor cascade designed by J. Sanzo For this case, the solution with the second-order extension of the Godunov method gives verygood agreement with the design distribution of parameters given by Sanzo
TL;DR: In this article, variable mesh difference methods of third order are derived for the solution of the two-point, second-order, singular perturbation problems y = f (x, y, y, e ).
1 Jan 1984
TL;DR: In this paper, the origins of the first scheme were discussed and the second and third schemes were executed on the intermediate layer and the final layer, respectively, and the first and second schemes on the third and fourth layers, respectively.
Abstract: 1. General Introduction.- 1.1 Introduction.- 1.2 Boundary Value Problems and Initial Problems.- 1.3 One-Dimensional Unsteady Flow Characteristics.- 1.4 Steady Supersonic Plane or Axi-Symmetric Flow. Equations of Motion in Characteristic Form.- 1.5 Basic Concepts Used in Finite Difference Methods.- References.- 2. The Godunov Schemes.- 2.1 The Origins of Godunov's First Scheme.- 2.2 Godunov's First Scheme. One-Dimensional Eulerian Equations.- 2.3 Godunov's First Scheme in Two and More Dimensions.- 2.4 Godunov's Second Scheme.- 2.5 The Double Sweep Method.- 2.6 Execution of the Second Scheme on the Intermediate Layer.- 2.7 Boundary Conditions on the Intermediate Layer.- 2.8 Procedure on the Final Layer.- 2.9 Applications of the Second Godunov Scheme.- 2.10 Glimm's Method.- 2.11 Outline of Solution for Gas Dynamic Equations.- 2.12 The Glimm Scheme for Simple Acoustic Waves.- 2.13 Random Choice for the Gas Dynamic Equations.- 2.14 Solution of the Riemann Problem.- 2.15 Extension to Unsteady Flow with Cylindrical or Spherical Symmetry.- 2.16 Remarks on Multi-Dimensional Problems.- References.- 3. The BVLR Method.- 3.1 Description of Method for Supersonic Flow.- 3.2 Extensions to Mixed Subsonic-Supersonic Flow. The Blunt Body Problem.- 3.3 The Double Sweep Method for Unsteady Three-Dimensional Flow.- 3.4 Worked Problem. Application to Circular Arc Airfoil.- 3.5 Results and Discussion.- Appendix-Shock Expansion Theory.- References.- 4. The Method of Characteristics for Three-Dimensional Problems in Gas Dynamics.- 4.1 Introduction.- 4.2 Bicharacteristics Method (Butler).- 4.3 Optimal Characteristics Methods (Bruhn and Haack, Schaetz).- 4.4 Near Characteristics Method (Sauer).- References.- 5. The Method of Integral Relations.- 5.1 Introduction.- 5.2 General Formulation. Model Problem.- 5.3 Flow Past Ellipses.- 5.4 The Supersonic Blunt Body Problem.- 5.5 Transonic Flow.- 5.6 Incompressible Laminar Boundary Layer Equations. Basic Formulation.- 5.7 The Method in the Compressible Case.- 5.8 Laminar Boundary Layers with Suction or Injection.- 5.9 Extension to Separated Flows.- 5.10 Application to Supersonic Wakes and Base Flows.- 5.11 Application to Three-Dimensional Laminar Boundary Layers.- 5.12 A Modified Form of the Method of Integral Relations.- 5.13 Application to Viscous Supersonic Conical Flows.- 5.14 Extension to Unsteady Laminar Boundary Layers.- 5.15 Application to Internal Flow Problems.- Model Problem (Chu and Gong).- References.- 6. Telenin's Method and the Method of Lines.- 6.1 Introduction.- 6.2 Solution of Laplace's Equation by Telenin's Method.- 6.3 Solution of a Model Mixed Type Equation by Telenin's Method.- 6.4 Application of Telenin's Method to the Symmetrical Blunt Body Problem.- 6.5 Extension to Unsymmetrical Blunt Body Flows.- 6.6 Application of Telenin's Method to the Supersonic Yawed Cone Problem.- 6.7 The Method of Lines. General Description.- 6.8 Applications of the Method of Lines.- 6.9 Powell's Method Applied to Two Point Boundary Value Problems.- Telenin's Method. Model Problems (Klopfer).- References.
TL;DR: In this article, an effective finite difference model has been developed to study the temperature profiles of the work roll and the strip, and the effect of changing velocities was also studied.
Abstract: The lack of a practical mathematical model to simulate thermal behavior of the metal rolling process has forced mill operators and designers to rely on plant experience and testing, which is time consuming and expensive. An effective finite difference model has been developed to study the temperature profiles of the work roll and the strip. Several finite difference techniques have been successfully employed to cope with the special characteristics of the rolling process, such as very high velocity, high temperature variation in a very thin layer, curved boundary, and bimaterial interface. Typical rolling conditions were analyzed to provide temperature information on the roll and strip. Both cold and hot rollings were considered, and the effect of changing velocities was also studied. Good correspondence is found when present results are compared with either analytical solutions under simplified rolling conditions or measured data.
TL;DR: In this article, the authors showed that the finite difference method with arbitrary irregular meshes for some class of elliptic problems converges linearly with the size of the star and confirmed the correctness of this theorem by numerical tests.
TL;DR: In this article, a non-orthogonal, algebraic coordinate transformation is used which yields a rectangular solution domain, which avoids the task of numerically generating boundary-fitted coordinates.
Abstract: A solution methodology has been developed for convection-diffusion problems in which one boundary of the solution domain does not lie along a coordinate line. A nonorthogonal, algebraic coordinate transformation is used which yields a rectangular solution domain. This transformation avoids the task of numerically generating boundary-fitted coordinates. The discretized conservation equations are derived on a control-volume basis. These equations contain pseudodiffusion terms that result from the nonorthogonal nature of the transformation. The entire discretization procedure is documented in detail. Although it is not an essential feature of the method, the discretized equations and their solutions are tied in with the well-documented practices of the Patankar solution scheme for orthogonal systems. Application of the methodology is illustrated by two numerical examples.
TL;DR: In this paper, two new classes of finite difference schemes are applied to the numerical solution of parabolic partial differential equations, and the stability of the new schemes is examined using a linear stability analysis and some numerical results are presented.
Abstract: Two new classes of finite difference schemes are applied to the numerical solution of parabolic partial differential equations. The formulae derived are self starting, are at least second order in time, are unconditionally stable and, unlike the Crank–Nicolson method, are $L^0$-stable in the sense of Gourlay and Morris. The stability of the new schemes is examined using a linear stability analysis and some numerical results are presented.
TL;DR: In this article, the main concern is the specification of boundary conditions to properly account for the effect of the wake on the rotor blade and the resulting computations of hovering and high advance ratio cases compare well with experiment.
Abstract: Rotary wing finite difference methods are investigated. The main concern is the specification of boundary conditions to properly account for the effect of the wake on the blade. Examples are given of an approach where wake effects are introduced by specifying an equivalent angle of attack. An alternate approach is also given where discrete vortices are introduced into the finite difference grid. The resulting computations of hovering and high advance ratio cases compare well with experiment. Some consideration is also given to the modeling of low to moderate advance ratio flows.
TL;DR: An alternating implicit differencing scheme (AIDS) was proposed in this paper for advancing the time-domain form of Maxwell's curl equations, which is a perturbation of the Crank-Nicolson method, and thus should have no stability limitation on?t.
Abstract: An alternating implicit differencing scheme (AIDS) is described for advancing the time-domain form of Maxwell's curl equations. This scheme is a perturbation of the Crank-Nicolson method, and thus should have no stability limitation on ?t. Test problems have been run with ?t incremented by factors of 10 up to 106 × the Courant-limit. In no case was any instability observed, although most runs yielded erroneous solutions if ?t was more than 103 × the Courant limit.
Book•
1 Oct 1984
TL;DR: In this paper, the right inverses of difference operators are proved for linear difference equations and nonlinear difference equations, and the existence proofs for right-inverse difference operators is established.
Abstract: Linear difference equations.- Existence proofs for right inverses of difference operators.- Nonlinear difference equations.
TL;DR: In this article, the accuracy and cost of three finite element methods for solving the linearized, two-dimensional shallow water equations are compared with a traditional explicit finite difference technique by comparing numerical and analytic plane wave solutions.
TL;DR: In this article, the authors derived results about the numerical performance of multi-point finite difference formulas for the differentiation of non-exact data and showed that such formulas can be constructed asymptotically unbiased and have a bounded amplification factor.
Abstract: In this paper, we derive results about the numerical performance of multi-point (moving average) finite difference formulas for the differentiation of non-exact data. In particular, we show that multi-point differentiators can be constructed which are asymptotically unbiased and have a bounded amplification factor as the steplength decreases and the number of points increases.
TL;DR: In this article, an algorithm for the solution of the three-dimensional resistive magnetohydrodynamic equations in toroidal geometry is presented, which employs the pseudospectral method for approximation in the two periodic coordinates, and finite differences in the radial direction.
TL;DR: The existence of weak discrete shocks for a wide class of difference approximations to systems of conservation laws is proved in this paper, where the difference schemes have to be conservative, kth order accurate, and, roughly speaking, (k + 1)th order dissipative, where k = 1 or 3.