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A Transformation Theory of the Partial Differential Equations of Gas Dynamics

Description: Note presenting a transformation theory of systems of partial differential equations which allows the construction of classes of pressure-density relations depending on parameters for which the equations governing the flow can be transformed into an essentially simpler form, into the system corresponding to the wave equation in the subsonic region, and finally into the form corresponding to the Tricomi equation in the transonic region.
Date: April 1950
Creator: Loewner, Charles
Partner: UNT Libraries Government Documents Department

Steepest descent for partial differential equations of mixed type

Description: The method of steepest descent is used to solve partial differential equations of mixed type. In the main hypothesis for this paper, H, L, and S are Hilbert spaces, T: H -> L and B: H -> S are functions with locally Lipshitz Fréchet derivatives where T represents a differential equation and B represents a boundary condition. Define ∅(u)=1/2IIT(u)II^2. Steepest descent is applied to the functional ∅. A new smoothing technique is developed and applied to Tricomi type equations (which are of mixed type).
Date: August 1992
Creator: Kim, Keehwan
Partner: UNT Libraries

Performance and scaling of locally-structured grid methods forpartial differential equations

Description: In this paper, we discuss some of the issues in obtaining high performance for block-structured adaptive mesh refinement software for partial differential equations. We show examples in which AMR scales to thousands of processors. We also discuss a number of metrics for performance and scalability that can provide a basis for understanding the advantages and disadvantages of this approach.
Date: July 19, 2007
Creator: Colella, Phillip; Bell, John; Keen, Noel; Ligocki, Terry; Lijewski, Michael & Van Straalen, Brian
Partner: UNT Libraries Government Documents Department

Seismic imaging of reservoir flow properties: Time-lapse pressurechanges

Description: Time-lapse fluid pressure and saturation estimates are sensitive to reservoir flow properties such as permeability. In fact, given time-lapse estimates of pressure and saturation changes, one may define a linear partial differential equation for permeability variations within the reservoir. The resulting linear inverse problem can be solved quite efficiently using sparse matrix techniques. An application to a set of crosswell saturation and pressure estimates from a CO{sub 2} flood at the Lost Hills field in California demonstrates the utility of this approach. From the crosswell estimates detailed estimates of reservoir permeability are produced. The resulting permeability estimates agree with a permeability log in an adjacent well and are in accordance with water and CO{sub 2} saturation changes in the interwell region.
Date: April 8, 2003
Creator: Vasco, Don W.
Partner: UNT Libraries Government Documents Department

Time-periodic solutions of the Benjamin-Ono equation

Description: We present a spectrally accurate numerical method for finding non-trivial time-periodic solutions of non-linear partial differential equations. The method is based on minimizing a functional (of the initial condition and the period) that is positive unless the solution is periodic, in which case it is zero. We solve an adjoint PDE to compute the gradient of this functional with respect to the initial condition. We include additional terms in the functional to specify the free parameters, which, in the case of the Benjamin-Ono equation, are the mean, a spatial phase, a temporal phase and the real part of one of the Fourier modes at t = 0. We use our method to study global paths of non-trivial time-periodic solutions connecting stationary and traveling waves of the Benjamin-Ono equation. As a starting guess for each path, we compute periodic solutions of the linearized problem by solving an infinite dimensional eigenvalue problem in closed form. We then use our numerical method to continue these solutions beyond the realm of linear theory until another traveling wave is reached (or until the solution blows up). By experimentation with data fitting, we identify the analytical form of the solutions on the path connecting the one-hump stationary solution to the two-hump traveling wave. We then derive exact formulas for these solutions by explicitly solving the system of ODE's governing the evolution of solitons using the ansatz suggested by the numerical simulations.
Date: April 1, 2008
Creator: Ambrose , D.M. & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Dimensional reduction as a tool for mesh refinement and trackingsingularities of PDEs

Description: We present a collection of algorithms which utilizedimensional reduction to perform mesh refinement and study possiblysingular solutions of time-dependent partial differential equations. Thealgorithms are inspired by constructions used in statistical mechanics toevaluate the properties of a system near a critical point. The firstalgorithm allows the accurate determination of the time of occurrence ofa possible singularity. The second algorithm is an adaptive meshrefinement scheme which can be used to approach efficiently the possiblesingularity. Finally, the third algorithm uses the second algorithm untilthe available resolution is exhausted (as we approach the possiblesingularity) and then switches to a dimensionally reduced model which,when accurate, can follow faithfully the solution beyond the time ofoccurrence of the purported singularity. An accurate dimensionallyreduced model should dissipate energy at the right rate. We construct twovariants of each algorithm. The first variant assumes that we have actualknowledge of the reduced model. The second variant assumes that we knowthe form of the reduced model, i.e., the terms appearing in the reducedmodel, but not necessarily their coefficients. In this case, we alsoprovide a way of determining the coefficients. We present numericalresults for the Burgers equation with zero and nonzero viscosity toillustrate the use of the algorithms.
Date: June 10, 2007
Creator: Stinis, Panagiotis
Partner: UNT Libraries Government Documents Department

International Conference on Multiscale Methods and Partial Differential Equations.

Description: The International Conference on Multiscale Methods and Partial Differential Equations (ICMMPDE for short) was held at IPAM, UCLA on August 26-27, 2005. The conference brought together researchers, students and practitioners with interest in the theoretical, computational and practical aspects of multiscale problems and related partial differential equations. The conference provided a forum to exchange and stimulate new ideas from different disciplines, and to formulate new challenging multiscale problems that will have impact in applications.
Date: December 12, 2006
Creator: Hou, Thomas
Partner: UNT Libraries Government Documents Department

A New Stabilized Nodal Integration Approach

Description: A new stabilized nodal integration scheme is proposed and implemented. In this work, focus is on the natural neighbor meshless interpolation schemes. The approach is a modification of the stabilized conforming nodal integration (SCNI) scheme and is shown to perform well in several benchmark problems.
Date: February 8, 2006
Creator: Puso, M; Zywicz, E & Chen, J S
Partner: UNT Libraries Government Documents Department

Final Report on Subcontract B591217: Multigrid Methods for Systems of PDEs

Description: Progress is summarized in the following areas of study: (1) Compatible relaxation; (2) Improving aggregation-based MG solver performance - variable cycle; (3) First Order System Least Squares (FOSLS) for LQCD; (4) Auxiliary space preconditioners; (5) Bootstrap algebraic multigrid; and (6) Practical applications of AMG and fast auxiliary space preconditioners.
Date: October 25, 2011
Creator: Xu, J; Brannick, J J & Zikatanov, L
Partner: UNT Libraries Government Documents Department

Operator-Based Preconditioning of Stiff Hyperbolic Systems

Description: We introduce an operator-based scheme for preconditioning stiff components encoun- tered in implicit methods for hyperbolic systems of partial differential equations posed on regular grids. The method is based on a directional splitting of the implicit operator, followed by a char- acteristic decomposition of the resulting directional parts. This approach allows for solution to any number of characteristic components, from the entire system to only the fastest, stiffness-inducing waves. We apply the preconditioning method to stiff hyperbolic systems arising in magnetohydro- dynamics and gas dynamics. We then present numerical results showing that this preconditioning scheme works well on problems where the underlying stiffness results from the interaction of fast transient waves with slowly-evolving dynamics, scales well to large problem sizes and numbers of processors, and allows for additional customization based on the specific problems under study.
Date: February 9, 2009
Creator: Daniel R. Reynolds, Ravi Samtaney, and Carol S. Woodward
Partner: UNT Libraries Government Documents Department

The analysis of a sparse grid stochastic collocation method for partial differential equations with high-dimensional random input data.

Description: This work describes the convergence analysis of a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems and, as such, the derived strong error estimates for the fully discrete solution are used to compare the computational efficiency of the proposed method with the Monte Carlo method. Numerical examples illustrate the theoretical results and are used to compare this approach with several others, including the standard Monte Carlo.
Date: December 1, 2007
Creator: Webster, Clayton; Tempone, Raul (Florida State University, Tallahassee, FL) & Nobile, Fabio (Politecnico di Milano, Italy)
Partner: UNT Libraries Government Documents Department

Overture: An Object-Oriented Framework for Overlapping Grid Applications

Description: The Overture framework is an object-oriented environment for solving partial differential equations on over-lapping grids. We describe some of the tools in Overture that can be used to generate grids and solve partial differential equations (PDEs). Overture contains a collection of C++ classes that can be used to write PDE solvers either at a high level or at a lower level for efficiency. There are also a number of tools provided with Overture that can be used with no programming effort. These tools include capabilities to: repair computer-aided-design (CAD) geometries and build global surface triangulations; generate surface and volume grids with hyperbolic grid generation; generate composite overlapping grids; generate hybrid (unstructured) grids; and solve particular PDEs such as the incompressible and compressible Navier-Stokes equations.
Date: April 4, 2002
Creator: Henshaw, W. D.
Partner: UNT Libraries Government Documents Department

Control and Analysis of a Single-Link Flexible Beam with Experimental Verification

Description: The objective of this report is to ascertain the general conditions for the avoidance and reduction of residual vibration in a flexible manipulator. Conventional manipulators usually have a 1.5 to 2-m reach, and their associated dynamic models typically are composed of lumped parameter elements; the major compliance emanates from the, drive trains because of torsional loading effects. The energy storage of the drive system is predominantly potential energy because of the low inertia in the drive tram; thus simple spring models have been adequate. A long-reach manipulator with a large aspect ratio (length to diameter) is a fundamentally different problem. Energy storage for this type of manipulator is distributive by nature because of the potential energy resulting from bending and the kinetic energy due to deflection rates. Instead of ordinary differential equations, partial differential equations are required to describe this system, making the analysis more difficult. The general flexibility problem associated with a distributive dynamic system, with specific emphasis on flexible manipulator, will be addressed in this report. Furthermore, three control schemes will be discussed and demonstrated on, a single flexible manipulator to determine their general merits.
Date: January 1, 1992
Creator: Jansen, J.F.
Partner: UNT Libraries Government Documents Department

Overture: The grid classes

Description: Overture is a library containing classes for grids, overlapping grid generation and the discretization and solution of PDEs on overlapping grids. This document describes the Overture grid classes, including classes for single grids and classes for collections of grids.
Date: January 1, 1997
Creator: Brislawn, K.; Brown, D.; Chesshire, G. & Henshaw, W.
Partner: UNT Libraries Government Documents Department

Dynamics of the Ginzburg-Landau equations of superconductivity

Description: This article is concerned with the dynamical properties of solutions of the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity. It is shown that the TDGL equations define a dynamical process when the applied magnetic field varies with time, and a dynamical system when the applied magnetic field is stationary. The dynamical system describes the large-time asymptotic behavior: Every solution of the TDGL equations is attracted to a set of stationary solutions, which are divergence free. These results are obtained in the {open_quotes}{phi} = -{omega}({gradient}{center_dot}A){close_quotes} gauge, which reduces to the standard {close_quotes}{phi} = -{gradient}{center_dot}A{close_quotes} gauge if {omega} = 1 and to the zero-electric potential gauge if {omega} = 0; the treatment captures both in a unified framework. This gauge forces the London gauge, {gradient}{center_dot}A = 0, for any stationary solution of the TDGL equations.
Date: August 1997
Creator: Fleckinger-Pelle, J.; Kaper, H. G. & Takac, P.
Partner: UNT Libraries Government Documents Department

Dispersive water waves in one and two dimensions

Description: This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). We derived and analyzed new shallow water equations for one-dimensional flows near the critical Froude number as well as related integrable systems of evolutionary nonlinear partial differential equations in one spatial dimension, while developing new directions for the mathematics underlying the integrability of these systems. In particular, we applied the spectrum generating equation method to create and study new integrable systems of nonlinear partial differential equations related to our integrable shallow water equations. We also investigated the solutions of these systems of equations on a periodic spatial domain by using methods from the complex algebraic geometry of Riemann surfaces. We developed certain aspects of the required mathematical tools in the course of this investigation, such as inverse scattering with degenerate potentials, asymptotic reduction of the angle representations, geometric singular perturbation theory, modulation theory and singularity tracking for completely integrable equations. We also studied equations that admit weak solutions, i.e., solutions with discontinuous derivatives in the form of comers or cusps, even though they are solutions of integrable models, a property that is often incorrectly assumed to imply smooth solution behavior. In related work, we derived new shallow water equations in two dimensions for an incompressible fluid with a free surface that is moving under the force of gravity. These equations provide an estimate of the long-time asymptotic effects of slowly varying bottom topography and weak hydrostatic imbalance on the vertically averaged horizontal velocity, and they describe the flow regime in which the Froude number is small -- much smaller even than the small aspect ratio of the shallow domain.
Date: August 1, 1997
Creator: Holm, D.D. & Camassa, R.A.
Partner: UNT Libraries Government Documents Department

Perturbation approach and the constant of motion for on-dimensional dynamical systems

Description: A perturbation technic is used to find the constant of motion of a one-dimensional autonomous system. The convergence of the method is discussed through some examples. In addition, the approach is extended to one-dimensional non-autonomous systems where some examples are given.
Date: November 1, 1991
Creator: Lopez, G.
Partner: UNT Libraries Government Documents Department

Numerical Methods for Stochastic Partial Differential Equations

Description: This is the final report of a Laboratory Directed Research and Development (LDRD) project at the Los Alamos National laboratory (LANL). The objectives of this proposal were (1) the development of methods for understanding and control of spacetime discretization errors in nonlinear stochastic partial differential equations, and (2) the development of new and improved practical numerical methods for the solutions of these equations. The authors have succeeded in establishing two methods for error control: the functional Fokker-Planck equation for calculating the time discretization error and the transfer integral method for calculating the spatial discretization error. In addition they have developed a new second-order stochastic algorithm for multiplicative noise applicable to the case of colored noises, and which requires only a single random sequence generation per time step. All of these results have been verified via high-resolution numerical simulations and have been successfully applied to physical test cases. They have also made substantial progress on a longstanding problem in the dynamics of unstable fluid interfaces in porous media. This work has lead to highly accurate quasi-analytic solutions of idealized versions of this problem. These may be of use in benchmarking numerical solutions of the full stochastic PDEs that govern real-world problems.
Date: July 8, 1999
Creator: Sharp, D.H.; Habib, S. & Mineev, M.B.
Partner: UNT Libraries Government Documents Department

Final report: Stochastic partial differential equations applied to the predictability of complex multiscale phenomena

Description: The objectives of this research remain as stated in our proposal of November 1997. We report on progress in the quantification of uncertainty and prediction, with applications to flow in porous media and to shock wave physics. The main strength of this work is an innovative theory for the quantification of uncertainty based on models for solution errors in numerical simulations. We also emphasize a deep connection to application communities, including those in DOE Laboratories.
Date: August 30, 2001
Creator: Glimm, James; Deng, Yuefan; Lindquist, W. Brent & Tangerman, Folkert
Partner: UNT Libraries Government Documents Department

Using automatic differentiation for second-order matrix-free methods in PDE-constrained optimization.

Description: Classical methods of constrained optimization are often based on the assumptions that projection onto the constraint manifold is routine but accessing second-derivative information is not. Both assumptions need revision for the application of optimization to systems constrained by partial differential equations, in the contemporary limit of millions of state variables and in the parallel setting. Large-scale PDE solvers are complex pieces of software that exploit detailed knowledge of architecture and application and cannot easily be modified to fit the interface requirements of a blackbox optimizer. Furthermore, in view of the expense of PDE analyses, optimization methods not using second derivatives may require too many iterations to be practical. For general problems, automatic differentiation is likely to be the most convenient means of exploiting second derivatives. We delineate a role for automatic differentiation in matrix-free optimization formulations involving Newton's method, in which little more storage is required than that for the analysis code alone.
Date: November 20, 2000
Creator: Hovland, P. D.; Keyes, D. E.; McInnes, L. C. & Samyono, W.
Partner: UNT Libraries Government Documents Department

Overture: an objectoriented framework for solving partial differential equations on overlapping grids

Description: The Overture framework is an object-oriented environment for solving partial differential equations in two and three space dimensions. It is a collection of C++ libraries that enables the use of finite difference and finite volume methods at a level that hides the details of the associated data structures. Overture can be used to solve problems in complicated, moving geometries using the method of overlapping grids. It merges geometry, grid generation, difference operators, boundary conditions, data-base access and graphics into an easy to use high level interface.
Date: September 22, 1998
Creator: Brown, D L; Henshaw, W D & Quinlan , D J
Partner: UNT Libraries Government Documents Department

Final Report: Symposium on Adaptive Methods for Partial Differential Equations

Description: OAK-B135 Final Report: Symposium on Adaptive Methods for Partial Differential Equations. Complex physical phenomena often include features that span a wide range of spatial and temporal scales. Accurate simulation of such phenomena can be difficult to obtain, and computations that are under-resolved can even exhibit spurious features. While it is possible to resolve small scale features by increasing the number of grid points, global grid refinement can quickly lead to problems that are intractable, even on the largest available computing facilities. These constraints are particularly severe for three dimensional problems that involve complex physics. One way to achieve the needed resolution is to refine the computational mesh locally, in only those regions where enhanced resolution is required. Adaptive solution methods concentrate computational effort in regions where it is most needed. These methods have been successfully applied to a wide variety of problems in computational science and engineering. Adaptive methods can be difficult to implement, prompting the development of tools and environments to facilitate their use. To ensure that the results of their efforts are useful, algorithm and tool developers must maintain close communication with application specialists. Conversely it remains difficult for application specialists who are unfamiliar with the methods to evaluate the trade-offs between the benefits of enhanced local resolution and the effort needed to implement an adaptive solution method.
Date: December 10, 1998
Creator: Pernice, M.; Johnson, C.R.; Smith, P.J. & Fogelson, A.
Partner: UNT Libraries Government Documents Department

Modeling mesoscopic phenomena in extended dynamical systems

Description: This is the final report of a three-year, Laboratory-Directed Research and Development project at the Los Alamos National Laboratory (LANL). We have obtained classes of nonlinear solutions on curved geometries that demonstrate a novel interplay between topology and geometric frustration relevant for nanoscale systems. We have analyzed the nature and stability of localized oscillatory nonlinear excitations (multi-phonon bound states) on discrete nonlinear chains, including demonstrations of successful perturbation theories, existence of quasiperiodic excitations, response to external statistical time-dependent fields and point impurities, robustness in the presence of quantum fluctuations, and effects of boundary conditions. We have demonstrated multi-timescale effects for nonlinear Schroedinger descriptions and shown the success of memory function approaches for going beyond these approximations. In addition we have developed a generalized rate-equation framework that allows analysis of the important creation/annihilation processes in driven nonlinear, nonequilibiium systems.
Date: August 1, 1997
Creator: Bishop, A.; Lomdahl, P.; Jensen, N.G.; Cai, D.S.; Mertenz, F.; Konno, Hidetoshi et al.
Partner: UNT Libraries Government Documents Department