Showing papers on "Finite difference method published in 1986"
TL;DR: In this article, a non-iterative method for handling the coupling of the implicitly discretised time-dependent fluid flow equations is described, based on the use of pressure and velocity as dependent variables and is hence applicable to both the compressible and incompressible versions of the transport equations.
4,019 citations
TL;DR: In this paper, a finite-difference method for modeling P-SV wave propagation in heterogeneous media is presented, which is an extension of the method I previously proposed for modeling SH-wave propagation by using velocity and stress in a discrete grid, where the stability condition and the P-wave phase velocity dispersion curve do not depend on the Poisson's ratio.
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
2,583 citations
TL;DR: The PISO algorithm as mentioned in this paper is a non-iterative method for solving the implicity discretised, time-dependent, fluid flow equations, which is applied in conjunction with a finite-volume technique employing a backward temporal difference scheme to the computation of compressible and incompressible flow cases.
500 citations
TL;DR: In contrast with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships that are entirely consistent with the multigroup diffusion equation as discussed by the authors, which most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis.
344 citations
TL;DR: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system based on the pseudocompressibility approach.
Abstract: An implicit, finite difference computer code has been developed to solve the incompressible Navier-Stokes equations in a three-dimensional curvilinear coordinate system. The pressure field solution is based on the pseudocompressibility approach in which a time derivative pressure term is introduced into the mass conservation equation. The solution procedure employs an implicit, approximate factorization scheme. The Reynolds Stresses, which are uncoupled from the implicit scheme, are lagged by one time step to facilitate implementing various levels of the turbulence model. Test problems for external and internal flows are computer and the results are compared with existing experimental data. The application of this technique for general three-dimensional problems is then demonstrated.
275 citations
TL;DR: In this paper, spectral methods (Fourier Galerkin, Fourier pseudospectral, Chebyshev Tau, Chebyhev collocation, spectral element) and standard finite differences are applied to solve the Burgers equation with small viscosity (v = 1 100 π ).
202 citations
TL;DR: In this paper, the authors describe the application of the finite-difference method in the time domain to the solution of 3D eigenvalue problems, where the equations are discretized in space and time, and steady state solutions are then obtained via Fourier transform.
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.
180 citations
TL;DR: In this paper, 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 is described.
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.
171 citations
TL;DR: A moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension using p-hierarchic finite elements for the temporal integration of the solution, the error estimate, and the mesh motion.
Abstract: We discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in $H^1 $. We present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement.We discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. We use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on our method are presented for several examples.
114 citations
TL;DR: A Galerkin finite element method with quadratic interpolation is employed in solving Poisson's equation to yield the electric potential solution in this article, and a backward difference method is utilized to compute the space charge density from the continuity equation.
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.
97 citations
TL;DR: In this paper, a space-marching finite-difference algorithm is developed to solve the nonlinear inverse heat conduction problem, which uses interior temperature measurements at future times to estimate the surface heat flux.
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.
TL;DR: This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly and their dependence on various model parameters of physical interest is studied.
Abstract: A functional differential equation which is nonlinear and involves forward and backward deviating arguments is solved numerically. The equation models conduction in a myelinated nerve axon in which the myelin completely insulates the membrane, so that the potential change jumps from node to node. The equation is of first order with boundary values given at t = +/- infinity. The problem is approximated via a difference scheme which solves the problem on a finite interval by utilizing an asymptotic representation at the endpoints, cubic interpolation and iterative techniques to approximate the delays, and a continuation method to start the procedure. The procedure is tested on a class of problems which are solvable analytically to access the scheme's accuracy and stability, then applied to the problem that models propagation in a myelinated axon. The solution's dependence on various model parameters of physical interest is studied. This is the first numerical study of myelinated nerve conduction in which the advance and delay terms are treated explicitly.
TL;DR: In this article, the diffusion current at a finite band electrode of length L and width W in a cell of finite width (W) but infinite length and depth is obtained by an integral equation method which offers a promising alternative to finite difference methods.
01 Mar 1986
TL;DR: An efficient finite difference calculation procedure for three-dimensional recirculating flows is presented in this article, which is based on a coupled solution of the threedimensional momentum and continuity equations in primitive variables by the multigrid technique.
Abstract: An efficient finite difference calculation procedure for three-dimensional recirculating flows is presented. The algorithm is based on a coupled solution of the three-dimensional momentum and continuity equations in primitive variables by the multigrid technique. A symmetrical coupled Gauss-Seidel technique is used for iterations and is observed to provide good rates of smoothing. Calculations have been made of the fluid motion in a three-dimensional cubic cavity with a moving top wall. The efficiency of the method is demonstrated by performing calculations at different Reynolds numbers with finite difference grids as large as 66 × 66 × 66 nodes. The CPU times and storage requirements for these calculations are observed to be very modest. The algorithm has the potential to be the basis for an efficient general-purpose calculation procedure for practical fluid flows.
TL;DR: In this paper, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number.
Abstract: In this study, the equations of the three-dimensional convective motion of an infinite Prandtl number fluid are solved in spherical geometry, for Rayleigh numbers up to 15 times the critical number. An iterative method is used to find stationary solutions. The spherical parts of the operators are treated using a Galerkin collocation method while the radial and time dependences are expressed using finite difference methods. A systematic search for stationary solutions has led to eight different stream patterns for a low Rayleigh number (1.28 times the critical number). They can be classified as: I) Axisymmetrical solutions, analogous to rolls in plane geometry. II) Solutions which have several ascending plumes within a large area of ascending current, and also several descending plumes within an area of descending current. This type of flow is analogous to bimodal circulation in plane geometry. III) Solutions characterized by isolated ascending (or descending) plumes separated from each other by a...
TL;DR: In this paper, 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.
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.
TL;DR: In this article, a linear discontinuous finite difference formulation to solve the diffusion equations in coarse mesh and few groups is developed, where 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.
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.
TL;DR: A piecewise linear finite element-based method of lines is presented for the numerical solution of coupled parabolic partial differential equations which model biological and physicochemical reaction-diffusion processes in one space dimension and shows that the method is efficient, that a posteriori estimates of the space discretization error are accurate, and that the adaptive procedure reliably controls the space Discretization Error.
Book•
01 Jan 1986
TL;DR: In this paper, the authors present a model for overland flow over impermeable planes using dimensionless hydrographs and derive peak flow charts from the shape and peak flows.
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
TL;DR: A comparative study of eight discretization schemes for the equations describing convection-diffusion transport phenomena is presented, and the quadratic upstream difference schemes are shown to be superior in accuracy to the others at all Peclet numbers, for the test cases considered.
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
TL;DR: A semi-implicit difference method of second order in space is introduced for the numerical solution of the Euler equations if the Mach number e is small, and the solutions are second-order accurate also in time.
TL;DR: In this article, an approximate equation has been proposed to clarify the rotational vibration behavior of power transmission helical gear pairs with comparatively narrow facewidth, based on the theoretical deflection solved by one of the authors using the finite difference method.
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.
01 Jan 1986
TL;DR: Alternating-direction-implicil (ADI) and ‘upwind’ directional difference explicit (DDE) numerical schemes for solving the vorticity-transport equation are compared.
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.
TL;DR: In this article, 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.
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.
TL;DR: The development and analysis of various aspects of this class of schemes will be given along with the motivations behind many of the choices and various acceleration and efficiency modifications such as matrix reduction, diagonalization and flux split schemes are presented.
TL;DR: In this article, the three-dimensional turbulent flow in a curved hydraulic turbine draft tube is studied numerically, and the analysis is based on the steady Reynolds-averaged Navier-Stokes equations closed with the k-E model.
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.
TL;DR: In this paper, a dissipative term was introduced to the conventional explicit finite difference schemes, and a class of new explicit finite-difference schemes which are conditionally stable, span two time levels and are O(k,h^2 )$ accurate were derived.
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.
TL;DR: In this paper, 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.
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