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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

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

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

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

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

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

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

Variational particle scheme for the porous medium equation and for the system of isentropic Euler equations

Description: We introduce variational particle schemes for the porous medium equation and the system of isentropic Euler equations in one space dimension. The methods are motivated by the interpretation of each of these partial differential equations as a 'steepest descent' on a suitable abstract manifold. We show that our methods capture very well the nonlinear features of the flows.
Date: December 10, 2008
Creator: Westdickenberg, Michael & Wilkening, Jon
Partner: UNT Libraries Government Documents Department

Some free boundary problems in potential flow regime usinga based level set method

Description: Recent advances in the field of fluid mechanics with moving fronts are linked to the use of Level Set Methods, a versatile mathematical technique to follow free boundaries which undergo topological changes. A challenging class of problems in this context are those related to the solution of a partial differential equation posed on a moving domain, in which the boundary condition for the PDE solver has to be obtained from a partial differential equation defined on the front. This is the case of potential flow models with moving boundaries. Moreover the fluid front will possibly be carrying some material substance which will diffuse in the front and be advected by the front velocity, as for example the use of surfactants to lower surface tension. We present a Level Set based methodology to embed this partial differential equations defined on the front in a complete Eulerian framework, fully avoiding the tracking of fluid particles and its known limitations. To show the advantages of this approach in the field of Fluid Mechanics we present in this work one particular application: the numerical approximation of a potential flow model to simulate the evolution and breaking of a solitary wave propagating over a slopping bottom and compare the level set based algorithm with previous front tracking models.
Date: December 9, 2008
Creator: Garzon, M.; Bobillo-Ares, N. & Sethian, J.A.
Partner: UNT Libraries Government Documents Department

Peridynamics with LAMMPS : a user guide.

Description: Peridynamics is a nonlocal extension of classical continuum mechanics. The discrete peridynamic model has the same computational structure as a molecular dynamics model. This document provides a brief overview of the peridynamic model of a continuum, then discusses how the peridynamic model is discretized within LAMMPS. An example problem is also included.
Date: November 1, 2011
Creator: Lehoucq, Richard B.; Silling, Stewart Andrew; Seleson, Pablo (University of Texas at Austin, Austin, TX); Plimpton, Steven James & Parks, Michael L.
Partner: UNT Libraries Government Documents Department

Dynamic crack initiation toughness : experiments and peridynamic modeling.

Description: This is a dissertation on research conducted studying the dynamic crack initiation toughness of a 4340 steel. Researchers have been conducting experimental testing of dynamic crack initiation toughness, K{sub Ic}, for many years, using many experimental techniques with vastly different trends in the results when reporting K{sub Ic} as a function of loading rate. The dissertation describes a novel experimental technique for measuring K{sub Ic} in metals using the Kolsky bar. The method borrows from improvements made in recent years in traditional Kolsky bar testing by using pulse shaping techniques to ensure a constant loading rate applied to the sample before crack initiation. Dynamic crack initiation measurements were reported on a 4340 steel at two different loading rates. The steel was shown to exhibit a rate dependence, with the recorded values of K{sub Ic} being much higher at the higher loading rate. Using the knowledge of this rate dependence as a motivation in attempting to model the fracture events, a viscoplastic constitutive model was implemented into a peridynamic computational mechanics code. Peridynamics is a newly developed theory in solid mechanics that replaces the classical partial differential equations of motion with integral-differential equations which do not require the existence of spatial derivatives in the displacement field. This allows for the straightforward modeling of unguided crack initiation and growth. To date, peridynamic implementations have used severely restricted constitutive models. This research represents the first implementation of a complex material model and its validation. After showing results comparing deformations to experimental Taylor anvil impact for the viscoplastic material model, a novel failure criterion is introduced to model the dynamic crack initiation toughness experiments. The failure model is based on an energy criterion and uses the K{sub Ic} values recorded experimentally as an input. The failure model is then validated against one class of ...
Date: October 1, 2009
Creator: Foster, John T.
Partner: UNT Libraries Government Documents Department

Numerical study of a matrix-free trust-region SQP method for equality constrained optimization.

Description: This is a companion publication to the paper 'A Matrix-Free Trust-Region SQP Algorithm for Equality Constrained Optimization' [11]. In [11], we develop and analyze a trust-region sequential quadratic programming (SQP) method that supports the matrix-free (iterative, in-exact) solution of linear systems. In this report, we document the numerical behavior of the algorithm applied to a variety of equality constrained optimization problems, with constraints given by partial differential equations (PDEs).
Date: December 1, 2011
Creator: Heinkenschloss, Matthias (Rice University, Houston, TX); Ridzal, Denis & Aguilo, Miguel Antonio
Partner: UNT Libraries Government Documents Department

Filtering Algebraic Multigrid and Adaptive Strategies

Description: Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically.
Date: January 31, 2006
Creator: Nagel, A; Falgout, R D & Wittum, G
Partner: UNT Libraries Government Documents Department

A Marker Method for the Solution of the Damped Burgers' Equatio

Description: A new method for the solution of the damped Burgers' equation is described. The marker method relies on the definition of a convective field associated with the underlying partial differential equation; the information about the approximate solution is associated with the response of an ensemble of markers to this convective field. Some key aspects of the method, such as the selection of the shape function and the initial loading, are discussed in some details. The marker method is applicable to a general class of nonlinear dispersive partial differential equations.
Date: November 1, 2005
Creator: Lewandowski, Jerome L.V.
Partner: UNT Libraries Government Documents Department

R-matrix for a geodesic flow associated with a new integrable peakon equation

Description: We use the r-matrix formulation to show the integrability of geodesic flow on an N-dimensional space with coordinates q k, with k = 1, ..., N, equipped with the co-metric g i j = e-141--431(2 - e-Iq1-q3l). This flow is generated by a symmetry of the integrable partial differential equation (pde) mt + um, + 3mux = 0, m = u - a2uZx( ai s a constant), which was recently proven to be completely integrable and possess peakon solutions by Degasperis, Holm and Hone. The isospectral eigenvalue problem associated with this integrable pde is used to find a new Lax representation for its N-peakon solution dynamics. By employing this Lax matrix we obtain the r-matrix for the integrable geodesic flow.
Date: January 1, 2001
Creator: Holm, Darryl D. & Qiao, Z. (Zhijun)
Partner: UNT Libraries Government Documents Department

Algorithm refinement for stochastic partial differential equations.

Description: A hybrid particle/continuum algorithm is formulated for Fickian diffusion in the fluctuating hydrodynamic limit. The particles are taken as independent random walkers; the fluctuating diffusion equation is solved by finite differences with deterministic and white-noise fluxes. At the interface between the particle and continuum computations the coupling is by flux matching, giving exact mass conservation. This methodology is an extension of Adaptive Mesh and Algorithm Refinement to stochastic partial differential equations. A variety of numerical experiments were performed for both steady and time-dependent scenarios. In all cases the mean and variance of density are captured correctly by the stochastic hybrid algorithm. For a non-stochastic version (i.e., using only deterministic continuum fluxes) the mean density is correct, but the variance is reduced except within the particle region, far from the interface. Extensions of the methodology to fluid mechanics applications are discussed.
Date: January 1, 2001
Creator: Alexander, F. J. (Francis J.); Garcia, Alejandro L., & Tartakovsky, D. M. (Daniel M.)
Partner: UNT Libraries Government Documents Department

A Note on Equations for Steady-State Optimal Landscapes

Description: Based on the optimality principle (that the global energy expenditure rate is at its minimum for a given landscape under steady state conditions) and calculus of variations, we have derived a group of partial differential equations for describing steady-state optimal landscapes without explicitly distinguishing between hillslopes and channel networks. Other than building on the well-established Mining's equation, this work does not rely on any empirical relationships (such as those relating hydraulic parameters to local slopes). Using additional constraints, we also theoretically demonstrate that steady-state water depth is a power function of local slope, which is consistent with field data.
Date: June 15, 2010
Creator: Liu, H.H.
Partner: UNT Libraries Government Documents Department

Optimization and geophysical inverse problems

Description: A fundamental part of geophysics is to make inferences about the interior of the earth on the basis of data collected at or near the surface of the earth. In almost all cases these measured data are only indirectly related to the properties of the earth that are of interest, so an inverse problem must be solved in order to obtain estimates of the physical properties within the earth. In February of 1999 the U.S. Department of Energy sponsored a workshop that was intended to examine the methods currently being used to solve geophysical inverse problems and to consider what new approaches should be explored in the future. The interdisciplinary area between inverse problems in geophysics and optimization methods in mathematics was specifically targeted as one where an interchange of ideas was likely to be fruitful. Thus about half of the participants were actively involved in solving geophysical inverse problems and about half were actively involved in research on general optimization methods. This report presents some of the topics that were explored at the workshop and the conclusions that were reached. In general, the objective of a geophysical inverse problem is to find an earth model, described by a set of physical parameters, that is consistent with the observational data. It is usually assumed that the forward problem, that of calculating simulated data for an earth model, is well enough understood so that reasonably accurate synthetic data can be generated for an arbitrary model. The inverse problem is then posed as an optimization problem, where the function to be optimized is variously called the objective function, misfit function, or fitness function. The objective function is typically some measure of the difference between observational data and synthetic data calculated for a trial model. However, because of incomplete and inaccurate data, the ...
Date: October 1, 2000
Creator: Barhen, J.; Berryman, J.G.; Borcea, L.; Dennis, J.; de Groot-Hedlin, C.; Gilbert, F. et al.
Partner: UNT Libraries Government Documents Department

Final Report: Symposium on Adaptive Methods for Partial Differential Equations

Description: 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 8, 1998
Creator: Pernice, Michael; Johnson, Christopher R.; Smith, Philip J. & Fogelson, Aaron
Partner: UNT Libraries Government Documents Department