Showing papers on "Finite difference published in 1989"

Numerical simulation of three-dimensional saturated flow in randomly heterogeneous porous media

Abstract: This paper presents a numerical method for simulating flow fields in a stochastic porous medium that satisfies locally the Darcy equation, and has each of its hydraulic parameters represented as one realization of a three-dimensional random field. These are generated by using the Turning Bands method. Our ultimate objective is to obtain statistically meaningful solutions in order to check and extend a series of approximate analytical results previously obtained by a spectral perturbation method (L. W. Gelhar and co-workers). We investigate the computational aspects of the problem in relation with stochastic concepts. The difficulty of the numerical problem arises from the random nature of the hydraulic conductivities, which implies that a very large discretized algebraic system must be solved. Indeed, a preliminary evaluation with the aid of scale analysis suggests that, in order to solve meaningful flow problems, the total number of nodes must be of the order of 106. This is due to the requirement that Δxi ≪ gli ≪ Li, where Δxi is the mesh size, λi is a typical correlation scale of the inputs, and Li is the size of the flow domain (i = 1, 2, 3). The optimum strategy for the solution of such a problem is discussed in relation with supercomputer capabilities. Briefly, the proposed discretization method is the seven-point finite differences scheme, and the proposed solution method is iterative, based on prior approximate factorization of the large coefficient matrix. Preliminary results obtained with grids on the order of one hundred thousand nodes are discussed for the case of steady saturated flow with highly variable, random conductivities.

Finite-difference technique for SH-waves in 2-D media using irregular grids-application to the seismic response problem

Abstract: SUMMARY The finite-difference method is applied to compute the seismic response of 2-D inhomogeneous structures for SH-waves. A technique is proposed which uses an irregular grid (a rectangular grid with varying grid spacing). A geological structure may be composed of blocks of media inside of which velocity and density vary linearly in horizontal and vertical directions. The technique allows better adjusted modelling of a medium and, in numerical examples presented, yields more efficient computations as compared to those with regular grids. The technique is tested through comparison with a discrete-wavenumber method. As an example, the seismic response of the sediment-filled Chusal Valley, Carm region, USSR, is computed. The numerical results are compared with observations.

Efficient cell-vertex multigrid scheme for the three-dimensional Navier-Stokes equations

Abstract: A cell-vertex scheme for the three-dimensional Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing with locally varying coefficients, a multigrid method and carefully controlled dissipative terms, very good convergence rates are obtained for two- and three-dimensional flows. Details of the acceleration techniques, which are important for convergence on meshes with high aspect-ratio cells, are discussed. Emphasis is put on the analysis of the stability properties of the implicit smoothing of the explicit residuals with coefficients, which depend on cell aspect ratios.

A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation

Abstract: A finite-difference solution is demonstrated for an inverse problem of determining a control function p(t) in the parabolic partial differential equation ut=uxx+pu+f(x,t), 0

Modern homotopy methods in optimization

Abstract: Probability-one homotopy methods are a class of algorithms for solving nonlinear systems of equations that are accurate, robust, and converge from an arbitrary starting point almost surely These new techniques have been successfully applied to solve Brouwer faced point problems, polynomial systems of equations, and discretizations of nonlinear two-point boundary value problems based on shooting, finite differences, collocation, and finite elements This paper summarizes the theory of globally convergent homotopy algorithms for unconstrained and constrained optimization, and gives some examples of actual application of homotopy techniques to engineering optimization problems

Numerical solution of the acoustic and elastic wave equations by a new rapid expansion method1

Abstract: We present a new rapid expansion method (REM) for the time integration of the acoustic wave equation and the equations of dynamic elasticity in two spatial dimensions. The method is applicable to spatial grid methods such as finite differences, finite elements or the Fourier method. It is based on a Chebyshev expansion of the formal solution to the appropriate wave equation written in operator form. The method yields machine accuracy yet it is faster than methods based on temporal differencing. Its disadvantages are that it does not apply to all types of material rheology, and it can also require much storage when many snapshots and time sections are desired. Comparisons between numerical and analytical solutions for simple acoustic and elastic problems demonstrate the high accuracy of the REM.

Three‐dimensional unsteady flow simulations: Alternative strategies for a volume‐averaged calculation

Abstract: SUMMARY This work builds on a SIMPLE-type code to produce two numerical codes of greatly improved speed and accuracy for solution of the Navier-Stokes equations. Both implicit and explicit codes employ an improved QUICK (quadratic upstream interpolation for convective kinematics) scheme to finite difference convective terms for non-uniform grids. The PRIME (update pressure implicit, momentum explicit) algorithm is used as the computational procedure for the implicit code. Use of both the ICCG (incomplete Cholesky decomposition, conjugate gradient) method and the MG (multigrid) technique to enhance solution execution speed is illustrated. While the implicit code is first-order in time, the explicit is second-order accurate. Two- and threedimensional forced convection and sidewall-heated natural convection flows in a cavity are chosen as test cases. Predictions with the new schemes show substantial computational savings and very good agreement when compared to previous simulations and experimental data.

Nonlinear filters for efficient shock computation

Abstract: A new type of methods for the numerical approximation of hyperbolic conservation laws with discontinuous solution is introduced. The methods are based on standard finite difference schemes. The difference solution is processed with a nonlinear conservation form filter at every time level to eliminate spurious oscillations near shocks. It is proved that the filter can control the total variation of the solution and also produce sharp discrete shocks. The method is simpler and faster than many other high resolution schemes for shock calculations. Numerical examples in one and two space dimensions are presented.

Finite element/finite difference propagation algorithm for integrated optical device

Abstract: A propagation algorithm based on finite elements and a finite difference discretisation of the scalar wave equation is investigated as an alternative to the beam propagation method. The new approach overcomes the assumption of low contrast media in the BPM and allows the propagation of arbitrary input fields in strongly guiding structures.

Transfer Functions for Efficient Calculation of Multidimensional Transient Heat Transfer

Abstract: Finite difference or finite element methods reduce transient multidimensional heat transfer problems into a set of first-order differential equations when thermal physical properties are time invariant and the heat transfer processes are linear. This paper presents a method for determining the exact solution to a set of first-order differential equations when the inputs are modeled by a continuous, piecewise linear curve. For long-time solutions, the method presented is more efficient than Euler, Crank-Nicolson, or other classical techniques.

Numerical solution of the incompressible Navier-Stokes equations for steady-state and time-dependent problems

Abstract: An algorithm for the solution of the incompressible Navier-Stokes equations in three-dimensional generalized curvilinear coordinates is presented. The algorithm can be used to compute both steady-state and time-dependent flow problems. The algorithm is based on the method of artificial compressibility and uses a higher-order flux-difference splitting technique for the convective terms and a second-order central difference for the viscous terms. The steady-state solution of flow through a square duct with a 90 deg bend is computed and the results are compared with experimental data. Good agreement is observed. A comparison with an analytically known exact solution is then performed to verify the time accuracy of the algorithm. Finally, the flow through an artificial heart configuration with moving boundaries is calculated and presented.

New Directions in Solitons and Nonlinear Periodic Waves: Polycnoidal Waves, Imbricated Solitons, Weakly Nonlocal Solitary Waves, and Numerical Boundary Value Algorithms

Abstract: Publisher Summary This chapter describes polycnoidal waves, periodic waves as exact imbricate series of solitons, and numerical boundary value algorithms for direct computation of solitons. Polycnoidal waves are the spatially periodic generalizations of multiple solitons. Their theory explores what happens when solitary waves are not solitary, but instead form periodic wave trains. The nonlinear Fourier transform and theta function formalism are both extensions of inverse scattering. In contrast, the Stokes series and the Newton/Fourier/continuation polyalgorithm are direct methods that bypass inverse scattering completely. Imbricate series is an offshoot of polycnoidal theory that has acquired a life of its own. Inverse scattering-soluble equations often allow an exact nonlinear superposition principle. Imbricate series show that each soliton may behave as an independent dynamical system even when very close to other solitary waves. Crude finite difference discretizations have been replaced by exponentially accurate pseudospectral algorithms. Truncated domains have been rendered obsolete by new spectral basis sets, such as rational Chebyshev functions, which compute on the infinite domain. The generalization of the concept of a solitary wave is also presented.

Numerical simulation of shear band formation in a frictional-dilational material

Abstract: The explicit, finite difference code FLAC is used to model shear band development in isotropic, elastic-plastic, Coulomb, non-associated, non-hardening materials. The code reproduces shear band inclinations which depend on both the angle of friction and the angle of dilation. Localization occurs at the yield point. Shear band width is sensitive to the size of the finite difference mesh but is controlled also by the magnitudes of the friction and dilation angles. The distribution of shear bands throughout the specimen is also influenced by the values of friction and, to a lesser extent, of dilation. There is no suggestion that numerical instability or numerical truncation errors are responsible for shear band nucleation.

Some estimates for finite difference approximations

Abstract: Some estimates for the approximation of optimal stochastic control problems by discrete time problems are obtained. In particular an estimate for the solutions of the continuous time versus the discrete time Hamilton–Jacobi–Bellman equations is given. The technique used is more analytic than probabilistic.

Improving the accuracy of central difference schemes

Abstract: General difference approximations to the fluid dynamic equations require an artificial viscosity in order to converge to a steady state. This artificial viscosity serves two purposes. One is to suppress high frequency noise which is not damped by the central differences. The second purpose is to introduce an entropy-like condition so that shocks can be captured. These viscosities need a coefficient to measure the amount of viscosity to be added. In the standard scheme, a scalar coefficient is used based on the spectral radius of the Jacobian of the convective flux. However, this can add too much viscosity to the slower waves. Hence, it is suggested that a matrix viscosity be used. This gives an appropriate viscosity for each wave component. With this matrix valued coefficient, the central difference scheme becomes closer to upwind biased methods.

A cell-vertex multigrid method for the Navier-Stokes equations

Abstract: A cell-vertex scheme for the Navier-Stokes equations, which is based on central difference approximations and Runge-Kutta time stepping, is described. Using local time stepping, implicit residual smoothing, a multigrid method, and carefully controlled artificial dissipative terms, very good convergence rates are obtained for a wide range of two- and three-dimensional flows over airfoils and wings. The accuracy of the code is examined by grid refinement studies and comparison with experimental data. For an accurate prediction of turbulent flows with strong separations, a modified version of the nonequilibrium turbulence model of Johnson and King is introduced, which is well suited for an implementation into three-dimensional Navier-Stokes codes. It is shown that the solutions for three-dimensional flows with strong separations can be dramatically improved, when a nonequilibrium model of turbulence is used.

Direct simulations of turbulent flow using finite-difference schemes

Abstract: A high-order accurate finite-difference approach is presented for calculating incompressible turbulent flow. The methods used include a kinetic energy conserving central difference scheme and an upwind difference scheme. The methods are evaluated in test cases for the evolution of small-amplitude disturbances and fully developed turbulent channel flow. It is suggested that the finite-difference approach can be applied to complex geometries more easilty than highly accurate spectral methods. It is concluded that the upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries.

Analysis of forward wide-angle light propagation in semiconductor rib waveguides and integrated-optic structures

Abstract: We analyse longitudinally varying semiconductor rib waveguides with a new nonparaxial wide-angle equation for unidirectional light propagation. We further develop a solution method involving multiplication of the incoming electric field by a series of unitary operators which are evaluated by splitstep fast Fourier transform and finite difference techniques. In the test case of a strongly guiding rib waveguide Yjunction, the estimated losses nearly coincide with those of standard Fresnel equation procedures. We also present a numerical analysis of an integrated-optic lens, demonstrating significant errors in previous results. Finally, we explicitly illustrate the increased accuracy of our new method in comparison to the Fresnel equation for a highly nonparaxial Gaussian beam.