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Multigrid methods with applications to reservoir simulation

Description: Multigrid methods are studied for solving elliptic partial differential equations. Focus is on parallel multigrid methods and their use for reservoir simulation. Multicolor Fourier analysis is used to analyze the behavior of standard multigrid methods for problems in one and two dimensions. Relation between multicolor and standard Fourier analysis is established. Multiple coarse grid methods for solving model problems in 1 and 2 dimensions are considered; at each coarse grid level we use more than one coarse grid to improve convergence. For a given Dirichlet problem, a related extended problem is first constructed; a purification procedure can be used to obtain Moore-Penrose solutions of the singular systems encountered. For solving anisotropic equations, semicoarsening and line smoothing techniques are used with multiple coarse grid methods to improve convergence. Two-level convergence factors are estimated using multicolor. In the case where each operator has the same stencil on each grid point on one level, exact multilevel convergence factors can be obtained. For solving partial differential equations with discontinuous coefficients, interpolation and restriction operators should include information about the equation coefficients. Matrix-dependent interpolation and restriction operators based on the Schur complement can be used in nonsymmetric cases. A semicoarsening multigrid solver with these operators is used in UTCOMP, a 3-D, multiphase, multicomponent, compositional reservoir simulator. The numerical experiments are carried out on different computing systems. Results indicate that the multigrid methods are promising.
Date: May 1, 1994
Creator: Xiao, Shengyou
Partner: UNT Libraries Government Documents Department

An adaptive level set method

Description: This thesis describes a new method for the numerical solution of partial differential equations of the parabolic type on an adaptively refined mesh in two or more spatial dimensions. The method is motivated and developed in the context of the level set formulation for the curvature dependent propagation of surfaces in three dimensions. In that setting, it realizes the multiple advantages of decreased computational effort, localized accuracy enhancement, and compatibility with problems containing a range of length scales.
Date: December 1, 1995
Creator: Milne, R. B.
Partner: UNT Libraries Government Documents Department

Efficient biased random bit generation for parallel processing

Description: A lattice gas automaton was implemented on a massively parallel machine (the BBN TC2000) and a vector supercomputer (the CRAY C90). The automaton models Burgers equation {rho}t + {rho}{rho}{sub x} = {nu}{rho}{sub xx} in 1 dimension. The lattice gas evolves by advecting and colliding pseudo-particles on a 1-dimensional, periodic grid. The specific rules for colliding particles are stochastic in nature and require the generation of many billions of random numbers to create the random bits necessary for the lattice gas. The goal of the thesis was to speed up the process of generating the random bits and thereby lessen the computational bottleneck of the automaton.
Date: September 28, 1994
Creator: Slone, D.M.
Partner: UNT Libraries Government Documents Department

Fast methods for static Hamilton-Jacobi Partial Differential Equations

Description: The authors develop a family of fast methods approximating the solution to a wide class of static Hamilton-Jacobi partial differential equations. These partial differential equations are considered in the context of control-theoretic and front-propagation problems. In general, to produce a numerical solution to such a problem, one has to solve a large system of coupled non-linear discretized equations. The techniques use partial information about the characteristic directions to de-couple the system. Previously known fast methods, available for isotropic problems, are discussed in detail. They introduce a family of new Ordered Upwinding Methods (OUM) for general (anisotropic) problems and prove convergence to the viscosity solution of the corresponding Hamilton-Jacobi partial differential equation. The hybrid methods introduced here are based on the analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems. The performance of the methods is analyzed and compared to that of several other numerical approaches to these problems. Computational experiments are performed using test problems from control theory, computational geometry and seismology.
Date: May 1, 2001
Creator: Vladimirsky, Alexander Boris
Partner: UNT Libraries Government Documents Department

Finite Element Solutions to Nonlinear Partial Differential Equations

Description: This paper develops a numerical algorithm that produces finite element solutions for a broad class of partial differential equations. The method is based on steepest descent methods in the Sobolev space H¹(Ω). Although the method may be applied in more general settings, we consider only differential equations that may be written as a first order quasi-linear system. The method is developed in a Hilbert space setting where strong convergence is established for part of the iteration. We also prove convergence for an inner iteration in the finite element setting. The method is demonstrated on Burger's equation and the Navier-Stokes equations as applied to the square cavity flow problem. Numerical evidence suggests that the accuracy of the method is second order,. A documented listing of the FORTRAN code for the Navier-Stokes equations is included.
Date: August 1981
Creator: Beasley, Craig J. (Craig Jackson)
Partner: UNT Libraries

Physical Motivation and Methods of Solution of Classical Partial Differential Equations

Description: We consider three classical equations that are important examples of parabolic, elliptic, and hyperbolic partial differential equations, namely, the heat equation, the Laplace's equation, and the wave equation. We derive them from physical principles, explore methods of finding solutions, and make observations about their applications.
Date: August 1995
Creator: Thompson, Jeremy R. (Jeremy Ray)
Partner: UNT Libraries

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

The Development and Application of Reactive Transport Modeling Techniques to Study Radionuclide Migration at Yucca Mountain, NV

Description: Yucca Mountain, Nevada has been chosen as a possible site for the first high level radioactive waste repository in the United States. As part of the site investigation studies, we need to make scientifically rigorous estimations of radionuclide migration in the event of a repository breach. Performance assessment models used to make these estimations are computationally intensive. We have developed two reactive transport modeling techniques to simulate radionuclide transport at Yucca Mountain: (1) the selective coupling approach applied to the convection-dispersion-reaction (CDR) model and (2) a reactive stream tube approach (RST). These models were designed to capture the important processes that influence radionuclide migration while being computationally efficient. The conventional method of modeling reactive transport models is to solve a coupled set of multi-dimensional partial differential equations for the relevant chemical components in the system. We have developed an iterative solution technique, denoted the selective coupling method, that represents a versatile alternative to traditional uncoupled iterative techniques and the filly coupled global implicit method. We show that selective coupling results in computational and memory savings relative to these approaches. We develop RST as an alternative to the CDR method for solving large two- or three-dimensional reactive transport simulations for cases in which one is interested in predicting the flux across a specific control plane. In the RST method, the multidimensional problem is reduced to a series of one-dimensional transport simulations along streamlines. The key assumption with RST is that mixing at the control plane approximates the transverse dispersion between streamlines. We compare the CDR and RST approaches for several scenarios that are relevant to the Yucca Mountain Project. For example, we apply the CDR and RST approaches to model an ongoing field experiment called the Unsaturated Zone Transport Test.
Date: September 1, 1999
Creator: Viswanathan, Hari Selvi
Partner: UNT Libraries Government Documents Department

Development and application of the quasi-potential transformation

Description: The quasi-potential transformation, based on the Kirchhoff transformation, reduces the equations governing mass-transfer in a steady-state, nonconvective electrolytic system into two independent parts. The geometry-specific part involves the solution of Laplace`s equation subject to the relevant boundary conditions. The system-specific part involves the solution of a set of coupled first-order, nonlinear, ordinary differential equations. We develop a theoretical basis for the quasi-potential transformation using potential theory. The major assumption on which the quasi-potential transformation is based is that the concentrations can be written as single-valued functions of the electrostatic potential. We see how the system-specific part of the calculation is developed. Boundary conditions are outlined, and the geometry-specific calculations for the disk and hemisphere electrodes are developed. We combine the system-specific calculations for the binary and acidic copper sulfate solutions with these geometry-specific calculations to obtain complete concentration profiles, potential distributions, and current density distributions for these systems. We also investigate the effect of migration on limiting currents.
Date: August 1, 1992
Creator: Pillay, B.
Partner: UNT Libraries Government Documents Department