Showing papers on "Finite difference method published in 1984"

SH-wave propagation in heterogeneous media; velocity-stress finite-difference method

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.

SH-wave propagation in heterogeneous media: velocity-stress finite-difference method

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 twolayered 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.

Numerical simulation of attenuated wavefields using a Padé approximant method

Abstract: Summary. Realistic anelastic attenuation laws are usually formulated as convolution operators, but this representation is intractable for time-domain synthetic seismogram methods such as the finite difference method. An approach based on Pade approximants provides a convenient, accurate reformulation of general anelastic laws in differential form. The resulting differential operators form a uniformly convergent sequence of increasing order in the time derivative, and all are shown to be causal, stable and dissi- pative. In the special case of frequency-independent Q, all required coefficients for the operators are obtained in closed form in terms of Legendre polynomials. Low-order approximants are surprisingly accurate. Finite-difference impulse responses for a plane wave in a constant-Q medium, calculated with the fifth-order convergent, are virtually indistinguishable from the exact solution. The formulation is easily generalized to non-scalar waves. Moreover, this method provides a framework for incorporating amplitude-dependent attenuation into numerical simulations.

The stability of explicit Euler time‐integration for certain finite difference approximations of the multi‐dimensional advection–diffusion equation

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.

A numerical method for ocean‐acoustic normal modes

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.

Boundary integral equation method for linear porous‐elasticity with applications to soil consolidation

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.

Computer Analysis of Dielectric Waveguides: A Finite-Difference Method

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.

Mathematical Analysis of Segregations in Continuously-cast Slabs

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.

The discrete wavenumber/finite element method for synthetic seismograms

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.

Numerical investigation of supercritical Taylor-vortex flow for a wide gap

Abstract: For a wide gap ( R 1 / R 2 = 0.5) and large aspect ratios L/d , axisymmetric Taylor-vortex flow has been observed in experiments up to very high supercritical Taylor (or Reynolds) numbers. This axisymmetric Taylor-vortex flow was investigated numerically by solving the Navier–Stokes equations using a very accurate (fourth-order in space) implicit finite-difference method. The high-order accuracy of the numerical method, in combination with large numbers of grid points used in the calculations, yielded accurate and reliable results for large supercritical Taylor numbers of up to 100 Ta c (or 10 Re c ). Prior to this study numerical solutions were reported up to only 16 Ta c . The emphasis of the present paper is placed upon displaying and elaborating the details of the flow field for large supercritical Taylor numbers. The flow field undergoes drastic changes as the Taylor number is increased from just supercritical to 100 Ta c . Spectral analysis (with respect to z ) of the flow variables indicates that the number of harmonics contributing substantially to the total solution increases sharply when the Taylor number is raised. The number of relevant harmonics is already unexpectedly high at moderate supercritical Ta . For larger Taylor numbers, the evolution of a jetlike or shocklike flow structure can be observed. In the axial plane, boundary layers develop along the inner and outer cylinder walls while the flow in the core region of the Taylor cells behaves in an increasingly inviscid manner.

The rational approximation to the acoustic wave equation with bottom interaction

Abstract: The rational‐linear approximation to the wave equation is a full‐wave approach to modeling range‐dependent ocean acoustic propagation with bottom interaction. It is a one‐way wave equation which gives an accurate treatment of high‐angle propagation to angles of about 40° with respect to the horizontal. Reflection from sound speed and density discontinuities is treated using the natural wave equation matching conditions. Bathymetry is allowed to vary in range. A tridiagonal implicit finite‐difference solution of this equation has been implemented. It has several advantages over the tridiagonal Crank–Nicholson solution of the parabolic equation. It more accurately models high angles of propagation, treats attenuation as a function of path length rather than range, and models range‐dependent bathymetry in a way that suits the form of the one‐way wave equation. The numerical methods are fourth‐order accurate in depth. The resulting implicit range step is still in the simple tridiagonal form.

Instability of difference models for hyperbolic initial boundary value problems

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.

Numerical investigations on discharging silos

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.

Application of the Godunov method and its second-order extension to cascade flow modeling

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

A Finite-difference Method for a Class of Singular Two-point Boundary-value Problems

Abstract: On considere une methode de difference finie a 3 points pour le probleme singulier a 2 points limites: y''+(2/x)y'+f(x,y)=0, 0

Finite Difference Techniques for Partial Differential Equations

Abstract: Publisher Summary Analytical methods of solving partial differential equations are usually restricted to linear cases with simple geometries and boundary conditions. The increasing availability of more and more powerful digital computers has made more common the use of numerical methods for solving such equations, in addition to non-linear equations with more complicated boundaries and boundary conditions. This chapter describes one particular method, the method of numerical finite differences. This method is based on the representation of the continuously defined function τ (x,y,z,t) and its derivatives in terms of values of an approximation τ defined at particular, discrete points called gridpoints. From the appropriate Taylor's series expansions of τ about such gridpoints, forward, backward, and central difference approximations to derivatives of τ can be developed to convert the given partial differential equation and its initial and boundary conditions to a set of linear algebraic equations linking the approximations τ defined at the gridpoints.

A Numerical Heat Transfer Analysis of Strip Rolling

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.

Finite-Difference Solutions of Convection-Diffusion Problems in Irregular Domains, Using a Nonorthogonal Coordinate Transformation

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.

Two New Finite Difference Schemes for Parabolic Equations

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.

Finite difference modeling of rotor flows including wake effects

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.