Showing papers on "Discretization published in 1989"
Book•
01 Jan 1989
TL;DR: Inverse Boundary Value Problems (IBV) as discussed by the authors, the heat equation is replaced by the Tikhonov regularization and regularization by Discretization (TBD) method.
Abstract: Normed Spaces.- Bounded and Compact Operators.- Riesz Theory.- Dual Systems and Fredholm Alternative.- Regularization in Dual Systems.- Potential Theory.- Singular Integral Equations.- Sobolev Spaces.- The Heat Equation.- Operator Approximations .-Degenerate Kernel Approximation.- Quadrature Methods.- Projection Methods.- Iterative Solution and Stability.- Equations of the First Kind.- Tikhonov Regularization.- Regularization by Discretization.- Inverse Boundary Value Problems.- References.- Index.
2,323 citations
Book•
12 Oct 1989
TL;DR: This book discusses problems, formulations, Algorithms, and other issues that have not been Considered in the area of discretization of the Discrete Equations, as well as some of the methods used in solving these problems.
Abstract: Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.
491 citations
TL;DR: Two a posteriori error estimators for the mini-element discretization of the Stokes equations are presented, based on a suitable evaluation of the residual of the finite element solution, which are globally upper and locally lower bounds for the error of the infinite element discretized.
Abstract: We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.
387 citations
01 Jan 1989
TL;DR: This work focuses on non-rectangular vertical discretization of water table layers with modifications for special cases and three-dimensional representation of confining layers.
Abstract: ! Introduction.... .2 Method...... ......3 Theory ......3 Modifications for special cases.. ....12 Non-rectangular vertical discretization..... .12 Water table layers. ....14 Quasi three-dimensional representation of confining layers 14 Boundary conditions and discharge points.. ........15 Backward tracking .....18 Limitations. .19 Limitations due to underlying assumptions of the method.. ..19 Limitations due to discretization effects 19 Limitations due to uncertainty in parameters and boundary conditions...... ........20 Modpath....... ........22 Organization and structure. .22 Flow system files .....22 Main data file. ......22 Stress package data files.... ..24 Cell-by-cell budget file .28 Opening flow system files 28 Output summary of flow field data.. ....29 Interactive input. .29 Name of file containing flow system files. ......29 Oulput mode. .....30 Starting location data.. ......31 Direction of tracking computation 34 Criteria for terminating path lines.. ......34 Mass balance check 35 Cell-by-cell data summary.. .35 Output. ...35 Execution without interactive input 36
313 citations
TL;DR: A new Navier-Stokes algorithm for use on unstructured triangular meshes is presented, which can be shown to be equivalent to a finite-volume approximation for regular equilateral triangular meshes.
Abstract: A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.
293 citations
TL;DR: 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 using the Turning Bands method.
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.
233 citations
TL;DR: An algorithm for calculating optimal operating strategies in a multi-reservoir hydroelectric system, which can take into account inflow stochasticity and does not require discretization of the state space is described.
228 citations
TL;DR: In this article, the reduced basis method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier-Stokes equations by finite element methods.
Abstract: The reduced basis method is a type of reduction method that can be used to solve large systems of nonlinear equations involving a parameter. In this work, the method is used in conjunction with a standard continuation technique to approximate the solution curve for the nonlinear equations resulting from discretizing the Navier–Stokes equations by finite–element methods. This paper demonstrates that the reduced basis method can be implemented to approximate efficiently solutions to incompressible viscous flows. Choices of basis vectors, issues concerning the implementation of the method, and numerical calculations are discussed. Two fluid flow calculations are considered, the driven cavity problem and flow over a forward facing step.
216 citations
TL;DR: Themudpack as discussed by the authors is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations, where the solution regions are rectangles in two dimensions and boxes in three dimensions.
175 citations
DOI•
01 Jan 1989
TL;DR: The thesis develops methods to solve discrete-time finite-state partially observable Markov decision processes and proves that the policy improvement step in iterative discretization procedure can be replaced by the approximation version of linear support algorithm.
Abstract: The thesis develops methods to solve discrete-time finite-state partially observable Markov decision processes. For the infinite horizon problem, only discounted reward case is considered.
For the finite horizon problem, two new algorithms are developed. The first algorithm is called the relaxed region algorithm. For each support in the value function, this algorithm determines a region not smaller than its support region and modifies it implicitly in later steps until the exact support region is found. The second algorithm, called linear support algorithm, systematically approximates the value function until all supports in the value function are found. The most important feature of this algorithm is that it can be modified to find an approximate value function. It has been shown that these two algorithms are more efficient than the one-pass algorithm.
For the infinite horizon problem, it is first shown that the approximation version of linear support algorithm can be used to substitute the policy improvement step in a standard successive approximation method to obtain an $\epsilon$-optimal value function. Next, an iterative discretization procedure is developed which uses a small number of states to find new supports and improve the value function between two policy improvement steps. Since only a finite number of states are chosen in this process, some techniques developed for finite MDP can be applied here. Finally, we prove that the policy improvement step in iterative discretization procedure can be replaced by the approximation version of linear support algorithm.
The last part of the thesis deals with problems with continuous signals. We first show that if the signal processes are uniformly distributed, then the problem can be reformulated as a problem with finite number of signals. Then the result is extended to where the signal processes are step functions. Since step functions can be easily used to approximate most of the probability distributions, this method can be used to approximate most of the problems with continuous signals. Finally, we present some conditions which guarantee that the linear support can be computed for any given state, then the methods developed for finite signal cases can be easily modified and applied to problems for which the conditions hold.
173 citations
TL;DR: A simple recursive iteration of the leapfrog discretization of Newton's equations leads to a removal of the finite-step-size error to any desired order in a manner that preserves phase-space areas and reversibility.
Abstract: We present a simple recursive iteration of the leapfrog discretization of Newton's equations which leads to a removal of the finite-step-size error to any desired order. This is done in a manner that preserves phase-space areas and reversibility, as required for use in the hybrid Monte Carlo method for simulating fermionic fields. The resulting asymptotic volume dependence is exp((ln/ital V/)/sup 1/2/). We test the scheme on the (2+1)-dimensional Hubbard model.
TL;DR: In this paper, it was shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation.
Abstract: It is shown that the discrete measures given by the Nanbu simulation method converge with respect to the weak topology of measures to solutions of the Boltzmann equation. The main conditions for this result are that the Cauchy problem for the Boltzmann equation has a sufficiently smooth solution and that the discretization parameters (cell size, timestep and test particle number) satisfy suitable constraints.
TL;DR: It is shown how a rather high performance can be achieved for the more effective preconditioners, such as successive over-relaxation and incomplete decompositions, on most vector computers if used in a straightforward manner.
Abstract: The discretization of second-order elliptic partial differential equations over three-dimensional rectangular regions, in general, leads to very large sparse linear systems. Because of their huge order and their sparseness, these systems can only be solved by iterative methods using powerful computers, e.g., vector supercomputers. Most of those methods are only attractive when used in combination with a so-called preconditioning matrix. Unfortunately, the more effective preconditioners, such as successive over-relaxation and incomplete decompositions, do not perform very well on most vector computers if used in a straightforward manner. In this paper it is shown how a rather high performance can be achieved for these preconditioners.
TL;DR: The cubic nonlinear Schr\"odinger equation and some of its discretizations, one of which is integrable, are studied and results regarding a forced nonlinear Schnitzer equation are briefly described.
Abstract: The cubic nonlinear Schr\"odinger equation and some of its discretizations, one of which is integ- rable, are studied. Apart from the integrable version the discretizations produce chaotic solutions for intermediate levels of mesh (mode) refinement. Chaos disappears when the discretization is fine enough and convergence to a quasiperiodic solution is obtained. Details are given for finite difference calculations, although similar results are also obtained by Fourier spectral methods. Results regarding a forced nonlinear Schr\"odinger equation are briefly described.
TL;DR: This paper shows by means of simple examples that problems can arise even for the simpler Stokes equations, and shows that care must be exercised in the choice of the pressure approximation for coupled flow problems.
Abstract: SUMMARY It is generally accepted that mixed and penalty finite element methods can routinely solve the incompressible Navier-Stokes equations. This paper shows by means of simple examples that problems can arise even for the simpler Stokes equations. The causes of the problem fall in either of two categories: round-off and ill conditioning, or a poor choice of pressure discretization. Nonsensical solutions can be obtained. Computation of the discrete divergence of the flow field is a simple and powerful tool to diagnose such conditions. In the first part of the paper several simple techniques for minimizing the effect of round-off are reviewed. In the second part it is shown that, for coupled flow problems, care must be exercised in the choice of the pressure approximation. A unified treatment of various observations by different workers is presented. This should prove useful for general users of the finite element method.
TL;DR: An estimate for the Hausdorff distance is presented, using an averaged modulus of continuity for multifunctions for multifunction, and a computational procedure to obtain a certain solution of the discretized inclusion is proposed.
Abstract: We present an estimate for the Hausdorff distance between the set of solutions of a differential inclusion and the set of solutions of its Euler discrete approximation, using an averaged modulus of continuity for multifunctions. A computational procedure to obtain a certain solution of the discretized inclusion is proposed.
TL;DR: In this article, Ahmed's vehicle-like body in ground proximity was modeled as a k-e model of turbulent flows and the Reynolds-averaged Navier-Stokes equations were solved numerically in a general curvilinear coordinate system for three-dimensional turbulent flows around Ahmed's body.
Abstract: The Reynolds-averaged Navier-Stokes equations, together with the equations of the k-e model of turbulence, were solved numerically in a general curvilinear coordinate system for three-dimensional turbulent flows around Ahmed's vehicle-like body. The numerical computations were performed for various afterbody upper-surface slant angles at a constant Reynolds number of 4.3 x 10 based on the body length. A multistep correction procedure was incorporated into the incompressibility condition, and the solution of the Poisson equation for pressure was obtained from the conjugate gradient method. A second-order discretization scheme for the convection term was applied to reduce the numerical diffusion. Most of the essential measured features of the flowfield around Ahmed's vehicle-like body in ground proximity, such as the formation of trailing vortices and the reverse flow region resulting from separation, were well predicted.
TL;DR: In this paper, a special discretization of the Dirichlet equation based on its geometric interpretation is proposed, and an iterative method that produces a monotonically convergent sequence is presented.
Abstract: The equation indicated in the title is the simplest representative of the class of nonlinear equations of Monge-Ampere type. Equations with such nonlinearities arise in dynamic meteorology, geometric optics, elasticity and differential geometry. In some special cases heuristic procedures for numerical solution are available, but in order for them to be successful a good initial guess is required. For a bounded convex domain, nonnegativef and Dirichlet data we consider a special discretization of the equation based on its geometric interpretation. For the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence. No special information about an initial guess is required, and to initiate the iterates a routine step is made. The method is self-correcting and is structurally suitable for a parallel computer. The computer program modules and several examples are presented in two appendices.
TL;DR: In this article, the authors studied continuous time collocation, time discretization and their global and discrete convergence properties in a mixed Volterra-Fredholm type of integral equations.
Abstract: Integral equations of mixed Volterra-Fredholm type arise in various physical and biological problems. In the present paper we study continuous time collocation, time discretization and their global and discrete convergence properties.
TL;DR: In this paper, a numerical method for the computation of heteroclinic orbits connecting two saddle points in ℝ2 is presented, which can be computed to very high period due to an integral phase condition and an adaptive discretization.
TL;DR: In this paper, the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers' equation is analyzed, and it is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth.
Abstract: The convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid Burgers’ equation is analyzed. It is proved that this kind of vanishing viscosity is responsible for spectral decay of those Fourier coefficients located toward the end of the computed spectrum; consequently, the discretization error is shown to be spectrally small, independently of whether or not the underlying solution is smooth. This in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments.
TL;DR: In this article, an iterative two-step procedure was proposed to incorporate multi-component interactions into one single two-dimensional finite difference transport code for equilibrium ion exchange during two dimensional transport.
TL;DR: Possible ways of developing accurate and efficient finite volume methods for complex, incompressible flows are discussed and examples of a finite volume method for general two-dimensional flows are presented.
TL;DR: It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used.
Abstract: It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used. Bounds for the Euclidean condition number as a function of the number of degrees f freedom are derived in $n \geq 2$ dimensions. When $n \geq 3$ the bound is the same as for the regular mesh case, but when $n = 2$ a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes. Applications of the results to the conjugate-gradient iterative method for solving such linear systems are given.
IBM1
TL;DR: It is seen that discretization may adversely affect the order of magnitude of this risk criterion, so care needs to be taken in practice to limit the degree of discretizations employed.
Abstract: The kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator Nevertheless, for computational (and especially graphical) reasons, exact kernel density estimates are often presented in (piecewise constant) discretized or (piecewise linear) interpolated form, using the exact estimate only at some grid of points The asymptotic integrated mean squared error properties of these modifications are studied It is seen that discretization may adversely affect the order of magnitude of this risk criterion, so care needs to be taken in practice to limit the degree of discretization employed This roughness effect essentially disappears when interpolation is used instead; then, it takes a remarkably coarse grid to result in more than a negligible deterioration in the estimate's performance Data discretization prior to applying the kernel density estimation prescription is also addressed Such prebinning does not affect the smoot
09 Jan 1989
TL;DR: In this paper, two algorithms for the solution of the time-dependent Euler equations are presented for unsteady aerodynamic analysis of oscillating airfoils for use on an unstructured grid made up of triangles.
Abstract: Two algorithms for the solution of the time-dependent Euler equations are presented for unsteady aerodynamic analysis of oscillating airfoils. Both algorithms were developed for use on an unstructured grid made up of triangles. The first flow solver involves a Runge-Kutta time-stepping scheme with a finite-volume spatial discretization that reduces to central differencing on a rectangular mesh. The second flow solver involves a modified Euler time-integration scheme with an upwind-biased spatial discretization based on the flux-vector splitting of Van Leer. The paper presents descriptions of the Euler solvers and dynamic mesh algorithm along with results which assess the capability.
TL;DR: In this paper, a finite-difference method with non-uniform discretization for the analysis of channel waveguides is presented, and the boundary conditions for either the quasi-TE or quasi-TM mode is illustrated.
Abstract: A finite-difference method (FDM) with nonuniform discretization for the analysis of channel waveguides is presented. Application of the boundary conditions for either the quasi-TE or quasi-TM mode is illustrated. Flexible discretization of the grid structures minimizes memory size, resulting in much smaller computing time without sacrificing the accuracy of the solution. This nonuniform discretization FDM technique is used to model the well-guided small-mode-size Ti:LiNbO/sub 3/ waveguides. The model treats both finite and infinite source diffusion cases. Quasi-TM mode profiles and the corresponding eigenvalues are rigorously evaluated and the theoretical results agree very well with the experimental results. >
TL;DR: A simple but powerful method for solving the transient stability problem with a high degree of parallelism is implemented and can significantly increase computational efficiency in a parallel-processing environment.
Abstract: A simple but powerful method for solving the transient stability problem with a high degree of parallelism is implemented. The transient stability is seen as a coupled set of nonlinear algebraic and differential equations. By applying a discretization method such as the trapezoidal rule, the overall algebraic-differential set of equations is transformed into a unique algebraic problem at each time step. A solution that considers every time step, not in a sequential way, but concurrently, is suggested. The solution of this set of equations with a relaxation-type indirect method gives rise to a highly parallel algorithm. The parallelism consists of a parallelism in space (that is in the equations at each time step) and a parallelism in time. Another characteristic of the algorithm is that the time step can be changed between iterations using a nested iteration multigrid technique from a coarse time grid to the desired fine time grid to enhance the convergence of the algorithm. The method has been tested on various size power systems, for various solution time periods, and various types of disturbances. It is shown that the method has good convergence properties and can significantly increase computational efficiency in a parallel-processing environment. >
TL;DR: In this article, the authors describe a continuum principle of design sensitivity analysis of linear and nonlinear structures for shape and nonshape optimization problems, which is more convenient to interpret and discretize for numerical calculations.