Abstract: I present a finite-difference method for modeling P-SV wave propagation in heterogeneous media This is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid The two components of the velocity cannot be defined at the same node for a complete staggered grid: the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio, while the S-wave phase velocity dispersion curve behavior is rather insensitive to the Poisson's ratio Therefore, the same code used for elastic media can be used for liquid media, where S-wave velocity goes to zero, and no special treatment is needed for a liquid-solid interface Typical physical phenomena arising with P-SV modeling, such as surface waves, are in agreement with analytical results The weathered-layer and corner-edge models show in seismograms the same converted phases obtained by previous authors This method gives stable results for step discontinuities, as shown for a liquid layer above an elastic half-space The head wave preserves the correct amplitude Finally, the corner-edge model illustrates a more complex geometry for the liquid-solid interface As the Poisson's ratio v increases from 025 to 05, the shear converted phases are removed from seismograms and from the time section of the wave field

Abstract: This paper describes the application of the finite-difference method in the time domain to the solution of three-dimensional (3-D) eigenvalue problems. Maxwell's equations are discretized in space and time, and steady-state solutions are then obtained via Fourier transform. While achieving the same accuracy and versatility as the TLM method, the finite-difference-time-domain (FD-TD) method requires less than half the CPU time and memory under identical simulation conditions. Other advantages over the TLM method include the absence of dielectric boundary errors in the treatment of 3-D inhomogeneous planar structures, such as microstrip. Some numerical results, including dispersion curves of a microstrip on anisotropic substrate, are presented.

Abstract: This paper describes the use of the MacCormack explicit time-spilitting scheme in the development of a two-dimensional (in plan) hydraulic simulation model that solves the St. Venant equations. Various tests devised to assess the performance of the method have been performed and the results are reported. Finally, two industrial applications of the model are presented. The method has been found to be computationally efficient and warrants further development.

Abstract: An accurate and efficient numerical scheme is presented for calculating electrical conditions inside wire‐duct electrostatic precipitators. A Galerkin finite‐element method with quadratic interpolation is employed in solving Poisson’s equation to yield the electric potential solution. A backward difference method is utilized to compute the space‐charge density from the continuity equation. The two methods are iteratively applied until convergence criteria for electric potential and current density are met. Computed potential and electric field values show good agreement with analytic solutions and experimental measurements. Comparisons between the present scheme and a finite‐difference scheme show that the finite‐element method offers distinct advantages in predicting the electrical characteristics of precipitators.

Abstract: A new space-marching finite-difference algorithm is developed to solve the nonlinear inverse heat conduction problem. This algorithm uses interior temperature measurements at future times to estimate the surface heat flux. The results of this method are compared on a test case with four other numerical schemes. The method is as accurate as the method developed by Beck [ 1] and uses a smaller computational time. This scheme is also employed to estimate the effects of different types of experimental errors on the estimation of the surface heat flux. Errors due to temperature measurements, thermocouple locations, and material properties are each investigated.

Abstract: A linear discontinuous finite difference formulation to solve the diffusion equations in coarse mesh and few groups is developed. The correction factors for heterogeneities, coarse mesh, and spectral effects are general interface flux discontinuity factors that can be explicitly calculated (synthetized) from detailed diffusion or transport solutions in fine mesh (heterogeneous) and multigroups, preserving the integrated fluxes and interface net currents. The stability is explicitly established for general synthetizations and for specific fine to coarse mesh and group reductions. Computing methods have been implemented for one-group (grey) synthetic diffusion acceleration, two-dimensional nodal/local solutions, and three-dimensional nodal simulation of pressurized water reactor cores. Results demonstrate the simplicity and stability of the formulation, a regular behaviour of the correction factors, an outstanding acceleration performance, and high potential for parallel and vector computing.

Abstract: An analytical technique for obtaining the time-resolved heat flux of a turbine blade is applied to the case of a TFE 731-2 hp full-stage rotating turbine. In order to obtain the heat flux values from the thin film gage temperature histories, a finite difference procedure is used to solve the heat equation with variable thermal properties. After setting out the data acquisition and analysis procedures, their application is illustrated for three midspan locations on the blade and operation at the design flow function. Results demonstrate that the magnitude of the heat flux fluctuation due to vane-balde interaction is large by comparison to the time-averaged heat flux at all investigated locations; FFT of a portion of the heat flux record illustrates that the dominant frequencies occur at the wake-cutting frequency and its harmonics.

Abstract: 1. Introduction. Historical review. Classical hydrology. Hydrodynamic equations. Infiltration. 2. Analysis of Runoff. Introduction. Dynamic equations. Simplified equations. The kinematic equations. Kinematic flow over impermeable planes. Friction equation. 3. Hydrograph Shape and Peak Flows. Design parameters. Solution of kinematic equations for flow off a plane. Hydrographs for planes. Derivation of peak flow charts. Effect of canalization. Estimation of abstractions. 4. Kinematic Assumptions. Nature of kinematic equations. Kinematic approximation to overland flow. Kinematic and non-kinematic waves. Non-kinematic waves. Muskingum river routing. Kinematic and diffusion models. 5. Numerical Solutions. Methods of solution of equations of motion. Method of characteristics. Finite difference methods. Numerical solution. Accuracy and stability of numerical schemes. Effect of friction. Choosing an explicit finite difference scheme for the solution of the one-dimensional kinematic equations. 6. Dimensionless Hydrographs. Unit hydrographs. Development and use of graphs. Excess rainfall. Dimensionless equations. Use of dimensionless hydrographs. 7. Storm Dynamics and Distribution. Design practice. Storm patterns. Numerical models. Solutions for dynamic storms. 8. Conduit Flow. Kinematic equations for non-rectangular sections. Part-full circular pipes. Computer program for design of storm drain network. Trapezoidal channels. Comparison of kinematic and time-shift routing in conduits. 9. Urban Catchment Management. Effects of urbanization. Example: Calculation of peak runoff for various conditions. Detention storage. Channel storage. Kinematic equations for closed conduit systems. Computer program to simulate reservoir level variations in a pipe network. 10. Kinematic Modelling. Introduction. Stormwater modelling. Mathematical models. System definition. Terminology and definitions. Modelling approaches. Examples of parametric and deterministic models. Two-dimensional overland flow modelling. 11. Applications of Kinematic Modelling. Approaches. A model for urban watersheds. A model for rural watersheds. Overland flow and streamflow program. Real-time modelling. 12. Groundwater Flow. General comments. Flow in porous media. Differential equations in porous media. Analysis of subsurface flow. Flow in unsaturated zone. Flow in non-homogenous saturated zone. Author Index. Index

Abstract: An approximate equation has been proposed to clarify the rotational vibration behaviour of power transmission helical gear pairs with comparatively narrow facewidth. It has been based on the theoretical deflection solved by one of the authors using the finite difference method. and the rotational vibration has been treated as a single degree of freedom system and the meshing resonance frequency of it has been obtained. Furthermore, its propriety is verified by measuring the acceleration for each gear pair belonging to the three categories classified by contact ratio. it is found that the meshing resonance frequencies calculated by use of the proposed equation agrees with experimental values.

Abstract: A comparative study of eight discretization schemes for the equations describing convection-diffusion transport phenomena is presented The (differencing) schemes considered are the conventional central, upwind and hybrid difference schemes,1,2 together with the quadratic upstream,3,4 quadratic upstream extended4 and quadratic upstream extended revised difference4 schemes Also tested are the so called locally exact difference scheme5 and the power difference scheme6 In multi-dimensional problems errors arise from ‘false diffusion’ and function approximations It is asserted that false diffusion is essentially a multi-dimensional source of error Hence errors associated with false diffusion may be investigated only via two- and three-dimensional problems The above schemes have been tested for both one- and two-dimensional flows with sources, to distinguish between ‘discretization’ errors and ‘false diffusion’ errors7 The one-dimensional study is reported in Reference 7 For 2D flows, the quadratic upstream difference schemes are shown to be superior in accuracy to the others at all Peclet numbers, for the test cases considered The stability of the schemes and their CPU time requirements are also discussed

Abstract: Separated flow past a circular cylinder is computed from two finite-difference Navier–Stokes models. Stream functions are calculated using a successive-over-relaxation (SOR) procedure. Alternating-direction-implicil (ADI) and ‘upwind’ directional difference explicit (DDE) numerical schemes for solving the vorticity-transport equation are compared. The ‘upwind’ differencing technique produces artificial viscosity which damps the wake and suppresses vortex shedding. It is shown to be unreliable and so the ADI approach is recommended.

Abstract: A variational, finite-difference method for computing the normalized propagation constants and the normalized field profiles of channel waveguides with arbitrary index profiles as well as aspect ratios is presented. Mode dispersion curves and the field profiles of the fundamental mode of channel waveguides having profiles of practical interest are included.

Abstract: SUMMARY The three-dimensional turbulent flow in a curved hydraulic turbine draft tube is studied numerically. The analysis is based on the steady Reynolds-averaged Navier-Stokes equations closed with the k--E model. The governing equations are discretized by a conservative finite volume formulation on a non-orthogonal bodyfitted co-ordinate system. Two grid systems, one with 34 x 16 x 12 nodes and another with 50 x 30 x 22 nodes, have been used and the results from them are compared. In terms of computing effort, the number of iterations needed to yield the same degree of convergence is found to be proportional to the square root of the total number of nodes employed, which is consistent with an earlier study made for two-dimensional flows using the same algorithm. Calculations have been performed over a wide range of inlet swirl, using both the hybrid and second-order upwind schemes on coarse and fine grids. The addition of inlet swirl is found to eliminate the stalling characteristics in the downstream region and modify the behaviour of the flow markedly in the elbow region, thereby affecting the overall pressure recovery noticeably. The recovery factor increases up to a swirl ratio of about 0.75, and then drops off. Although the general trends obtained with both finite difference operators are in agreement, the quantitative values as well as some of the fine flow structures can differ. Many of the detailed features observed on the fine grid system are smeared out on the coarse grid system, pointing out the necessity of both a good finite difference operator and a good grid distribution for an accurate result.

Abstract: Most conventional explicit finite difference schemes, e.g. Euler’s scheme, for solving the parabolic equation of Schrodinger type $u_t = iu_{xx} $ are unconditionally unstable. This difficulty can be overcome by introducing a dissipative term to the conventional explicit schemes. Based on this approach, we derive a class of new explicit finite difference schemes which are conditionally stable, spans two time levels and are $O(k,h^2 )$ accurate. We also determine the schemes from this class that have the least restrictive stability requirements. It is interesting to note that the analog of the Lax–Wendroff scheme is unstable.

Abstract: Numerical finite difference calculations for diffusional mass transport incorporating reactions between minerals and a fluid phase, based on a continuum representation of porous media, are compared with the exact solution. The finite difference algorithm is based on the weak formulation of the moving boundary problem in which a fixed grid of node points is used. Mineral reactions are considered to be in local equilibrium with a fluid phase and may take place either at sharp reaction fronts or distributed homogeneously throughout a control volume. The theory is applied to one- and two-component systems. Analytical solutions to the finite difference equations for the first and second node points provide for a detailed comparison with the exact solution. It was found that on a time scale that is small compared with the time required for the solid to completely dissolve at a single node point, the finite difference approximation yields a spurious behavior for the concentration and solid phase volume fraction. However, the finite difference algorithm reproduces the average behavior of the motion of the reaction front and concentration of the reacting species provided the advance of the front is sufficiently slow resulting in a quasi-steady state solution. A numerical example ismore » presented for the dissolution of quartz at 550 /sup 0/C and 1000 bars to illustrate the general theory.« less

Abstract: Distorted grid or ''corner point geometry'' have received increased attention as a means of enhancing the modelling flexibility of finite difference type reservoir simulators. Non-rectangular grids permit better alignment with reservoir boundaries and faults and may lead to fewer grid blocks and therefore less computational efforts in simulation of complex reservoirs. This paper shows that caution should be used as regards the transmissibility calculations for distorted grids. Specifically, it is demonstrated by simulation experiments that methods which try to maintain (for 2-D problems) the five-point connectivity of standard finite differences may lead to serious errors. A generalized nine-point scheme is constructed and shown to be more appropriate for distorted grids. For instance, the scheme seems to preserve the accuracy of a square nine-point scheme if the distortion is kept within certain limits. Although based on finite element techniques, the scheme is given a finite difference type form, and can therefore easily be implemented in more general-purpose finite difference type reservoir simulators.

Abstract: The paper studies with finite difference nonlinear circulation models the uncertainties in interesting flow properties, such as western boundary current transport, potential and kinetic energy, owing to the uncertainty in the driving surface boundary condition. The procedure is based upon nonlinear optimization methods. The same calculations permit quantitative study of the importance of new information as a function of type, region of measurement and accuracy, providing a method to study various observing strategies. Uncertainty in a model parameter, the bottom friction coefficient, is studied in conjunction with uncertain measurements. The model is free to adjust the bottom friction coefficient such that an objective function is minimized while fitting a set of data to within prescribed bounds. The relative importance of the accuracy of the knowledge about the friction coefficient with respect to various kinds of observations is then quantified, and the possible range of the friction coefficients is calculated.

Abstract: A novel approach to the solution of three-dimensional micro-wave cavity and waveguide problems by finite elements reduces the number of spurious, nonphysical modes. Solutions are obtained in terms of the field vector H. Three-dimensional vector boundary conditions are implemented in a way that allows arbitrarily-shaped curved boundaries to be modeled. The formulation is based on a subparametric finite element with 27 interpolation nodes.

Abstract: : A numerical method has been developed to represent unsteady boundary layers with large flow reversal. It makes use of the characteristic box scheme which examines the finite-difference grid in relation to the magnitude and direction of local velocity and reaches and implements a decision to ensure that the Raetz principle of regions of influence and dependence is not violated. The method has been applied to the problem of an impulsively started circular cylinder and the results though generally consistent with those of van Dommelen and Shen obtained with a Lagrangian method, show some differences. The time step is identified as very important and, with the present intelligent numerical scheme, the results were readily extended to times far beyond those previously achieved with Eulerian methods. Extrapolation of the results suggests that the much discussed singularity for this unsteady flow is the same as that of the corresponding steady flow.

Abstract: A computer method has been developed which uses the time domain finite-difference (TDFD) algorithm to calculate the deposition of the electromagnetic (EM) field in three-dimensional biological models. This, the first of two papers, describes the algorithm and the computer programs developed. The method is demonstrated by calculating the penetration of the EM field from a rectangular waveguide radiating into a homogeneous model, the calculation being carried out in two dimensions for simplicity in this paper.