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Operator overloading as an enabling technology for automatic differentiation

Description: We present an example of the science that is enabled by object-oriented programming techniques. Scientific computation often needs derivatives for solving nonlinear systems such as those arising in many PDE algorithms, optimization, parameter identification, stiff ordinary differential equations, or sensitivity analysis. Automatic differentiation computes derivatives accurately and efficiently by applying the chain rule to each arithmetic operation or elementary function. Operator overloading enables the techniques of either the forward or the reverse mode of automatic differentiation to be applied to real-world scientific problems. We illustrate automatic differentiation with an example drawn from a model of unsaturated flow in a porous medium. The problem arises from planning for the long-term storage of radioactive waste.
Date: May 1, 1993
Creator: Corliss, G. F. & Griewank, A.
Partner: UNT Libraries Government Documents Department

Numerical methods for molecular dynamics. Progress report

Description: This report summarizes our research progress to date on the use of multigrid methods for three-dimensional elliptic partial differential equations, with particular emphasis on application to the Poisson-Boltzmann equation of molecular biophysics. This research is motivated by the need for fast and accurate numerical solution techniques for three-dimensional problems arising in physics and engineering. In many applications these problems must be solved repeatedly, and the extremely large number of discrete unknowns required to accurately approximate solutions to partial differential equations in three-dimensional regions necessitates the use of efficient solution methods. This situation makes clear the importance of developing methods which are of optimal order (or nearly so), meaning that the number of operations required to solve the discrete problem is on the order of the number of discrete unknowns. Multigrid methods are generally regarded as being in this class of methods, and are in fact provably optimal order for an increasingly large class of problems. The fundamental goal of this research is to develop a fast and accurate numerical technique, based on multi-level principles, for the solutions of the Poisson-Boltzmann equation of molecular biophysics and similar equations occurring in other applications. An outline of the report is as follows. We first present some background material, followed by a survey of the literature on the use of multigrid methods for solving problems similar to the Poisson-Boltzmann equation. A short description of the software we have developed so far is then given, and numerical results are discussed. Finally, our research plans for the coming year are presented.
Date: December 31, 1991
Creator: Skeel, R. D.
Partner: UNT Libraries Government Documents Department

SIAM conference on applications of dynamical systems. Abstracts and author index

Description: A conference (Oct.15--19, 1992, Snowbird, Utah; sponsored by SIAM (Society for Industrial and Applied Mathematics) Activity Group on Dynamical Systems) was held that highlighted recent developments in applied dynamical systems. The main lectures and minisymposia covered theory about chaotic motion, applications in high energy physics and heart fibrillations, turbulent motion, Henon map and attractor, integrable problems in classical physics, pattern formation in chemical reactions, etc. The conference fostered an exchange between mathematicians working on theoretical issues of modern dynamical systems and applied scientists. This two-part document contains abstracts, conference program, and an author index.
Date: December 31, 1992
Partner: UNT Libraries Government Documents Department

Maximizing the value of thermally integrated hydroelectric generating facilities

Description: This paper presents a demonstration of a phenomenon known as hydro-shifting, which relates to the provision of two sources of electric power to customers. The extent that these resources are used in each period is shown to depend partly on the time of day in which the power is provided. Although the dispatcher could combine thermal resales with hydro power in equal proportions for its customers in both periods, a rationalization is presented that justifies the withholding of some water until the peak period, with the balance of off-peak demand satisfied in larger proportions by resales of thermal power. Value is optimized for electrical generation, and environmental externalities are incorporated as minimum or maximum flow limitations. This paper also presents evidence that the optimal degree of hydro-shifting depends on the size of the reservoir adjacent to the dam, with this shifting behavior being more pronounced for larger reservoirs.
Date: July 1, 1992
Creator: Edwards, B. K.; Flaim, S. J. & Ancrile, J. D.
Partner: UNT Libraries Government Documents Department

Automatic, self-adaptive control of unfold transformations

Description: I describe an automated approach to partial evaluation based on elementary simplifications and implemented by program transformations. The approach emphasizes program algebra and relies on canonical forms and distributive laws to expose instances to which the elementary simplifications apply. I discuss some of the considerations that led to the design of this approach. This design discussion should be useful both in understanding the structure of the partial evaluation transformations themselves, and as an example of how to approach the design of automated program transformations in general.This approach to partial evaluation has been applied to a number of practical examples of moderate complexity, including eliminating a data structure from a specification of practical interest, a partial-differential-equation solver. This approach has the virtues of being implemented, automated, and able to partially evaluate specifications of practical interest.
Date: June 1, 1994
Creator: Boyle, J. M.
Partner: UNT Libraries Government Documents Department

Application of multiquadric method for numerical solution of elliptic partial differential equations

Description: We have used the multiquadric (MQ) approximation scheme for the solution of elliptic partial differential equations with Dirichlet and/or Neumann boundary conditions. The scheme has the advantage to use the data points in arbitrary locations with an arbitrary ordering. Two dimensional Laplace, Poisson and Biharmonic equations describing the various physical processes, have been taken as the test examples. The agreement is found to be very good between the computed and exact solutions. The method also provides an excellent approximation with curve boundary.
Date: January 1, 1994
Creator: Sharan, M.; Kansa, E. J. & Gupta, S.
Partner: UNT Libraries Government Documents Department

An Electromagnetic Finite Difference Time Domain Analog Treatment of Small Signal Acoustic Interactions

Description: Hyperbolic partial differential equations encompass an extremely important set of physical phenomena including electromagnetics and acoustics. Small amplitude acoustic interactions behave much the same as electromagnetic interactions for longitudinal acoustic waves because of the similar nature of the governing hyperbolic equations. Differences appear when transverse acoustic waves are considered, nonetheless the strong analogy between the acoustic and electromagnetic phenomena prompted the development of a Finite Difference Time Domain (FDTD) acoustic analog to the existing electromagnetic FDTD technique. The advantage of an acoustic FDTD (AFDTD) code are as follows: (1) Boundary condition-free treatment of the acoustic scatterer -- only the intrinsic properties of the scatterer`s material are needed, no shell treatment or other set of special equations describing the macroscopic behavior of a sheet of material or a junction, etc. are required; this allows completely general geometries and materials in the model. (2) Advanced outer radiation boundary condition analogs -- in the electromagnetics arena, highly absorbing outer radiation boundary conditions have been developed that can be applied with little modification to the acoustics arena with equal success. (3) A suite of preexisting capabilities related to electromagnetic modeling -- this includes automated model generation and interaction visualization as its most important components and is best exemplified by the capabilities of the LLNL generated TSAR electromagnetic FDTD code.
Date: March 25, 1994
Creator: Kunz, Karl; Steich, David; Lewis, Kent; Landrum, Charles & Barth, Marvin
Partner: UNT Libraries Government Documents Department

The method of Laplace Transform MultiQuadrics (LTMQ) for the solution of the groundwater flow equation

Description: This paper presents a new numerical method, the Laplace Transform MultiQuadrics (LTMQ) method, developed for the solution of the diffusion-type parabolic Partial Differential Equation (PDE) of fluid flow through porous media. LTMQ combines a MultiQuadrics (MQ) approximation scheme with a Laplace transform formulation. The use of MQ in the spatial approximations allows the accurate description of problems in complex porous media with a very limited number of nodes. The Laplace transform formulation eliminates the need for time discretization, thus allowing an unlimited time step size without any loss of accuracy. LTMQ is tested against results from three test problems of groundwater flow obtained from a standard Finite Difference (FD) model, as well as from analytical solutions. An excellent agreement between the LTMQ and the analytical and FD solutions is observed, while significant reductions in computer execution times may be achieved.
Date: June 1, 1992
Creator: Moridis, G. J. & Kansa, E.
Partner: UNT Libraries Government Documents Department

Errors when shock waves interact due to numerical shock width

Description: A simple test problem proposed by Noh, a strong shock reflecting from a rigid wall, demonstrates a generic problem with numerical shock capturing algorithms at boundaries that Noh called ``excess wall heating.`` We show that the same type of numerical error occurs in general when shock waves interact. The underlying cause is the non-uniform convergence to the hyperbolic solution of the inviscid limit of the solution to the PDEs with viscosity. The error can be understood from an analysis of the asymptotic solution. For a propagating shock, there is a difference in the total energy of the parabolic wave relative to the hyperbolic shock. Moreover, the relative energy depends on the strength of the shock. The error when shock waves interact is due to the difference in the relative energies between the incoming and outgoing shock waves. It is analogous to a phase shift in a scattering matrix. A conservative differencing scheme correctly describes the Hugoniot jump conditions for a steady propagating shock. Therefore, the error from the asymptotics occurs in the transient when the waves interact. The entropy error that occurs in the interaction region remains localized but does not dissipate. A scaling argument shows that as the viscosity coefficient goes to zero, the error shrinks in spatial extend but is constant in magnitude. Noh`s problem of the reflection of a shock from a rigid wall is equivalent to the symmetric impact of two shock waves of the opposite family. The asymptotic argument shows that the same type of numerical error would occur when the shocks are of unequal strength. Thus, Noh`s problem is indicative of a numerical error that occurs when shocks interact due to the numerical shock width.
Date: March 4, 1993
Creator: Menikoff, R.
Partner: UNT Libraries Government Documents Department

Nonlinear Partial Differential Equations Invariant to a One-Parameter Family of Stretching Groups

Description: Nonlinear partial differential equations (PDEs) in one dependent and two independent variables (call them c, z, and t) occur in many technological applications. Typical PDEs and the contexts in which they arise are the following: c{sub t} = (c{sup n}){sub zz}, which occurs in plasma physics, hydrology, gas flow in porous media, and applied superconductivity; cc{sub t} = c{sub zz}, which describes the expulsion of fluid from a long, slender, heated pipe; c{sub t} = (c{sub z} {sup 13}){sub z}, which describes heat transport in turbulent superfluid He-II; and c{sub tt} = (c{sub zz/2}) {integral} {sub o}{sup 1}c{sub z}{sup 2} dz, which describes the motion of a shock-loaded elastic membrane. All of these equations are invariant to a one-parameter family of one-parameter stretching groups of the form c{prime} = {lambda}{sup a}c, t{prime} = {lambda}{sup {beta}}t, z{prime} = {lambda}z, 0 < {lambda} < {infinity} where {lambda} is the group parameter that labels the individual transformations of a group and {alpha} and {beta} are the parameters that label groups of the family. The parameters {alpha} and {beta} are connected by a linear relation Ma + N{beta} = L where M, N, and L are numbers determined by the structure of the PDE. Similarity solutions are of the PDE that are invariant to one group of the family, say, that for which {alpha} = {alpha}* and {beta} = {beta}*. Such solutions have the form c = t{sup {alpha}*/{beta}*} y(z/t{sup 1/{beta}*}) where y is a function of the single variable x = z/t{sup 1{beta}*}. When substituted into the PDE yields an ordinary differential equation for the function y(x).
Date: June 1, 1994
Creator: Dresner, L.
Partner: UNT Libraries Government Documents Department

A theoretical analysis of vertical flow equilibrium

Description: The assumption of Vertical Flow Equilibrium (VFE) and of parallel flow conditions, in general, is often applied to the modeling of flow and displacement in natural porous media. However, the methodology for the development of the various models is rather intuitive, and no rigorous method is currently available. In this paper, we develop an asymptotic theory using as parameter the variable R{sub L} = (L/H){radical}(k{sub V})/(k{sub H}). It is rigorously shown that present models represent the leading order term of an asymptotic expansion with respect to 1/R{sub L}{sup 2}. Although this was numerically suspected, it is the first time that is is theoretically proved. Based on the general formulation, a series of models are subsequently obtained. In the absence of strong gravity effects, they generalize previous works by Zapata and Lake (1981), Yokoyama and Lake (1981) and Lake and Hirasaki (1981), on immiscible and miscible displacements. In the limit of gravity-segregated flow, we prove conditions for the fluids to be segregated and derive the Dupuit and Dietz (1953) approximations. Finally, we also discuss effects of capillarity and transverse dispersion.
Date: January 1, 1992
Creator: Yortsos, Y. C.
Partner: UNT Libraries Government Documents Department

Improved deterministic calculational methods for irregularly shaped shields. Final report, September 30, 1988--November 30, 1990

Description: A new discrete nodal transport method has been developed for general two-dimensional curvilinear geometry by using a boundary-fitted coordinate transformation from the general `physical` coordinates to square `computational` coordinates. The metrics which appear in the transformed transport equation are expanded using a simple polynomial function, and the angular divergence term is treated in the same way it is treated in S{sub N} methods for curved geometries. Because the metrics of the transformation depend upon the computational coordinates, the technical details of the formal development of the nodal method differ from those of ordinary nodal methods for rectangular geometry. However, the computational process in the transformed rectangular coordinate system is very similar to that used in conventional discrete nodal transport methods. A discrete S{sub N} method also has been developed to solve the boundary-fitted coordinate transformed transport equation. Simple test problems for non-simple geometries were solved using the zeroth-order nodal method, the first-order nodal method, and the S{sub N} method for the same physical and computational grids. The results for the test problems studied showed that, for most performance criteria, the computational efficiency of the zeroth-order nodal method was the highest of the three methods.
Date: December 1, 1992
Creator: Dorning, J. J.
Partner: UNT Libraries Government Documents Department

Glass mixing theory and tracer study results from the SF-10 run

Description: A general, partial differential equation governing glass mixing in the Slurry Fed Ceramic Melter (SFCM) was derived and a solution obtained based upon certain simplifying assumptions. Tracer studies were then conducted in the SFCM during the SF-10 run to test the theory and characterize glass mixing in this melter. Analysis of the tracer data shows that glass mixing in the SFCM can be explained by use of a model of two, well-mixed tanks in series.
Date: August 1, 1988
Creator: Bowman, B. W. & Routt, K. R.
Partner: UNT Libraries Government Documents Department

Applied analysis/computational mathematics. Final report 1993

Description: This is the final report for the Courant Mathematics and Computing Laboratory (CMCL) research program for the years 1991--1993. Our research efforts encompass the formulation of physical problems in terms of mathematical models (both old and new), the mathematical analysis of such models, and their numerical resolution. This last step involves the development and implementation of efficient methods for large scale computation. Our analytic and numerical work often go hand in hand; new theoretical approaches often have numerical counterparts, while numerical experimentation often suggests avenues for analytical investigation.
Date: December 1, 1993
Creator: Lax, P. & Berger, M.
Partner: UNT Libraries Government Documents Department

The geometry of weak solutions of certain integrable nonlinar PDE`s

Description: We investigate the geometry of new classes of soliton-like weak solutions for integrable nonlinear equations. One example is the class of peakons introduced by Camassa and Holm [1993] for their integrable shallow water equation. Alber, Camassa, Holm and Marsden [1994a] put this shallow water equation into the framework of complex integrable Hamiltonian systems on Riemann surfaces and use special limiting procedures to obtain new solutions such as quasiperiodic solutions, n-solitons, solitons with quasiperiodic background, billiard, and n-peakon solutions and complex angle representations for them. They also obtain explicit formulas for phase shifts of interacting soliton solutions using the method of asymptotic reduction of the corresponding angle representations. The method they use for the shallow water equation also leads to a link between one of the members of the Dym hierarchy and geodesic flow on N-dimensional quadrics. Amongst these geodesics, particularly interesting ones are the umbilic geodesics, which generate the class of umbilic soliton solutions. Umbilic solitons have the property that as the space variable x tends to infinity, the solution tends to a periodic wave, and as x tends to minus infinity, it tends to the same periodic wave with a phase shift. Elliptic billiards may be obtained from the problem of geodesics on quadrics by collapsing along the shortest semiaxis. The corresponding Hamiltonian billiard flows axe associated to new classes of solutions of equations in the Dym hierarchy. Such billiard type solutions have discontinuous spatial derivative and, thus, are weak solutions for this class of PDE`s.
Date: December 31, 1994
Creator: Alder, M. S.; Camassa, R.; Holm, D. D. & Marsden, J. E.
Partner: UNT Libraries Government Documents Department

Intergenerational transfers and the social discount rate

Description: This paper investigates the relationship between intergenerational asset transfers and the choice of the discount rate for use in cost-benefit analysis in a model of a competitive overlapping generations economy constrained by a socially managed exhaustible resource. Provided that there are no distortions in capital markets and that all agents hold perfect foresight, cost-benefit techniques will result in a Pareto efficient resource allocation if the discount rate is set equal to the market rate of interest. But since the path of the interest rate depends on the level of intergenerational transfers, cost-benefit techniques do not ensure a socially desirable distribution of welfare between generations; a social optimum will result only if intergenerational transfers are properly chosen and enforced. Decentralized private altruism may result in intergenerational transfers that both present and future individuals would agree are too small if members of the present generation attach positive weight to the general welfare of future generations, not simply their personal descendants. In a world where intergenerational transfers are non-optimal, second-best policy-making may imply a constrained optimum that is inefficient. Together, these findings suggest that cost-benefit analysis is at best a partial criterion to policy formulation that should be used only in conjunction with ethical principles that define the proper distribution of welfare between present and future generations.
Date: August 1, 1992
Creator: Howarth, R. B. & Norgaard, R. B.
Partner: UNT Libraries Government Documents Department

The use of wavelet transforms in the solution of two-phase flow problems

Description: In this paper we present the use of wavelets to solve the nonlinear Partial Differential.Equation (PDE) of two-phase flow in one dimension. The wavelet transforms allow a drastically different approach in the discretization of space. In contrast to the traditional trigonometric basis functions, wavelets approximate a function not by cancellation but by placement of wavelets at appropriate locations. When an abrupt chance, such as a shock wave or a spike, occurs in a function, only local coefficients in a wavelet approximation will be affected. The unique feature of wavelets is their Multi-Resolution Analysis (MRA) property, which allows seamless investigational any spatial resolution. The use of wavelets is tested in the solution of the one-dimensional Buckley-Leverett problem against analytical solutions and solutions obtained from standard numerical models. Two classes of wavelet bases (Daubechies and Chui-Wang) and two methods (Galerkin and collocation) are investigated. We determine that the Chui-Wang, wavelets and a collocation method provide the optimum wavelet solution for this type of problem. Increasing the resolution level improves the accuracy of the solution, but the order of the basis function seems to be far less important. Our results indicate that wavelet transforms are an effective and accurate method which does not suffer from oscillations or numerical smearing in the presence of steep fronts.
Date: October 1, 1994
Creator: Moridis, G. J.; Nikolaou, M. & You, Yong
Partner: UNT Libraries Government Documents Department

Scalable, extensible, and portable numerical libraries

Description: Designing a scalable and portable numerical library requires consideration of many factors, including choice of parallel communication technology, data structures, and user interfaces. The PETSc library (Portable Extensible Tools for Scientific computing) makes use of modern software technology to provide a flexible and portable implementation. This talk will discuss the use of a meta-communication layer (allowing the user to choose different transport layers such as MPI, p4, pvm, or vendor-specific libraries) for portability, an aggressive data-structure-neutral implementation that minimizes dependence on particular data structures (even vectors), permitting the library to adapt to the user rather than the other way around, and the separation of implementation language from user-interface language. Examples are presented.
Date: January 1, 1995
Creator: Gropp, W. & Smith, B.
Partner: UNT Libraries Government Documents Department

Dynamic stability of maglev systems

Description: Because dynamic instability is not acceptable for any commercial maglev systems, it is important to consider this phenomenon in the development of all maglev systems. This study considers the stability of maglev systems based on experimental data, scoping calculations, and simple mathematical models. Divergence and flutter are obtained for coupled vibration of a three-degree-of-freedom maglev vehicle on a guideway consisting of double L-shaped aluminum segments attached to a rotating wheel. The theory and analysis developed in this study identifies basic stability characteristics and future research needs of maglev systems.
Date: April 1, 1992
Creator: Cai, Y.; Chen, S. S.; Mulcahy, T. M. & Rote, D. M.
Partner: UNT Libraries Government Documents Department

Proceedings: Sisal `93

Description: This report contain papers on: Programmability and performance issues; The case of an iterative partial differential equation solver; Implementing the kernal of the Australian Region Weather Prediction Model in Sisal; Even and quarter-even prime length symmetric FFTs and their Sisal Implementations; Top-down thread generation for Sisal; Overlapping communications and computations on NUMA architechtures; Compiling technique based on dataflow analysis for funtional programming language Valid; Copy elimination for true multidimensional arrays in Sisal 2.0; Increasing parallelism for an optimization that reduces copying in IF2 graphs; Caching in on Sisal; Cache performance of Sisal Vs. FORTRAN; FFT algorithms on a shared-memory multiprocessor; A parallel implementation of nonnumeric search problems in Sisal; Computer vision algorithms in Sisal; Compilation of Sisal for a high-performance data driven vector processor; Sisal on distributed memory machines; A virtual shared addressing system for distributed memory Sisal; Developing a high-performance FFT algorithm in Sisal for a vector supercomputer; Implementation issues for IF2 on a static data-flow architechture; and Systematic control of parallelism in array-based data-flow computation. Selected papers have been indexed separately for inclusion in the Energy Science and Technology Database.
Date: October 1, 1993
Creator: Feo, J. T.
Partner: UNT Libraries Government Documents Department

Nonlinear difference approximations for evolutionary PDEs

Description: The authors describe a procedure to improve both the accuracy and computational efficiency of finite difference schemes used to simulate nonlinear PDEs. The underlying idea is that of enslaving, which is the estimation of the small unresolved scales in terms of the larger resolved scales. They discuss details of the procedure and illustrate them in the context of the forced Burgers` equation in one dimension. They present computational examples that demonstrate the predicted increases in accuracy and efficiency.
Date: January 1, 1995
Creator: Jones, D. A.; Margolin, L. G. & Poje, A. C.
Partner: UNT Libraries Government Documents Department

On some general properties of parabolic conservation equations

Description: This report deals with certain general properties of partial differential equations of the form S(c)c{sub t} + q{sub z} = Q(c), where t may thought of as time, z as distance, c as an intensive quantity (e.g., temperature), and q its flux (e.g., heat flux), and where q depends on both c and c{sub z}. Six topics are studied, namely: Maximum and minimum principles; ordering of solutions; invariance to stretching (affine) groups; stability of steady states; comparability of solutions; and traveling wave solutions. Illustrative examples are given from the field of nonlinear diffusion, applied superconductivity, and helium cryogenics.
Date: October 1, 1993
Creator: Dresner, L.
Partner: UNT Libraries Government Documents Department

A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems

Description: A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.
Date: September 24, 1990
Creator: Elton, A.B.H.
Partner: UNT Libraries Government Documents Department