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At least 127 records · Page 7

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NASA NTRS1996

Numerical Algorithms Based on Biorthogonal Wavelets

Wavelet bases are used to generate spaces of approximation for the resolution of bidimensional elliptic and parabolic problems. Under some specific hypotheses relating the properties of the wavelets to the order of the involved operators, it is shown that an approximate solution can be built. This approximation is then stable and converges towards the exact solution. It is designed such that fast algorithms involving biorthogonal multi resolution analyses can be used to resolve the corresponding numerical problems. Detailed algorithms are provided as well as the results of numerical tests on partial differential equations defined on the bidimensional torus.

Ponenti, Pj.↗

NASA NTRS1996

Numerical Algorithms Based on Biorthogonal Wavelets

Ponenti, Pj.↗

NASA NTRS2005

A Study of Multigrid Preconditioners Using Eigensystem Analysis

The convergence properties of numerical schemes for partial differential equations are studied by examining the eigensystem of the discrete operator. This method of analysis is very general, and allows the effects of boundary conditions and grid nonuniformities to be examined directly. Algorithms for the Laplace equation and a two equation model hyperbolic system are examined.

Roberts, Thomas W.↗

NASA NTRS1999

A Textbook for a First Course in Computational Fluid Dynamics

This paper describes and discusses the textbook, Fundamentals of Computational Fluid Dynamics by Lomax, Pulliam, and Zingg, which is intended for a graduate level first course in computational fluid dynamics. This textbook emphasizes fundamental concepts in developing, analyzing, and understanding numerical methods for the partial differential equations governing the physics of fluid flow. Its underlying philosophy is that the theory of linear algebra and the attendant eigenanalysis of linear systems provides a mathematical framework to describe and unify most numerical methods in common use in the field of fluid dynamics. Two linear model equations, the linear convection and diffusion equations, are used to illustrate concepts throughout. Emphasis is on the semi-discrete approach, in which the governing partial differential equations (PDE's) are reduced to systems of ordinary differential equations (ODE's) through a discretization of the spatial derivatives. The ordinary differential equations are then reduced to ordinary difference equations (O(Delta)E's) using a time-marching method. This methodology, using the progression from PDE through ODE's to O(Delta)E's, together with the use of the eigensystems of tridiagonal matrices and the theory of O(Delta)E's, gives the book its distinctiveness and provides a sound basis for a deep understanding of fundamental concepts in computational fluid dynamics.

Zingg, D. W.↗

NASA NTRS1971

Convergence of the discretization equations which arise in stochastic control

A problem is considered which arises in the numerical analysis of the degenerate partial differential equations of stochastic control theory. The relationship between diffusions and finite state Markov chains is considered. The role of probability theory in numerical analysis is illustrated.

Kushner, H. J.↗

DOE OSTI2020

An energy-based coupling approach to nonlocal interface problems.

Nonlocal models provide accurate representations of physical phenomena ranging from fracture mechanics to complex subsurface flows, settings in which traditional partial differential equation models fail to capture effects caused by long-range forces at the microscale and mesoscale. However, the application of nonlocal models to problems involving interfaces, such as multimaterial simulations and fluid-structure interaction, is hampered by the lack of a physically consistent interface theory which is needed to support numerical developments and, among other features, reduces to classical models in the limit as the extent of nonlocal interactions vanish. In this paper, we use an energy-based approach to develop a formulation of a nonlocal interface problem which provides a physically consistent extension of the classical perfect interface formulation for partial differential equations. Numerical examples in one and two dimensions validate the proposed framework and demonstrate the scope of our theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1986

Solution of elliptic partial differential equations by fast Poisson solvers using a local relaxation factor. 2: Two-step method

A two-step semidirect procedure is developed to accelerate the one-step procedure described in NASA TP-2529. For a set of constant coefficient model problems, the acceleration factor increases from 1 to 2 as the one-step procedure convergence rate decreases from + infinity to 0. It is also shown numerically that the two-step procedure can substantially accelerate the convergence of the numerical solution of many partial differential equations (PDE's) with variable coefficients.

Chang, S. C.↗

NASA NTRS1983

Finite difference numerical methods for boundary control problems governed by hyperbolic partial differential equations

This paper briefly reviews convergent finite difference schemes for hyperbolic initial boundary value problems and their applications to boundary control systems of hyperbolic type which arise in the modelling of vibrations. These difference schemes are combined with the primal and the dual approaches to compute the optimal control in the unconstrained case, as well as the case when the control is subject to inequality constraints. Some of the preliminary numerical results are also presented.

Chen, G.↗

NASA NTRS1973

Direct numerical solution of three-dimensional equations containing elliptic operators.

A direct three-dimensional elliptic solver is presented for application in a wide class of numerical methods for solving partial differential equations in physics and engineering. The derived algorithm and FORTRAN code implement Buzbee, Golub and Nielson's proposed extension of Buneman's Cyclic-Reduction Poisson solver to three dimensions. Both a 'most direct' cyclic reduction and a revised method (to eliminate roundoff error difficulties) are derived. Tests on an IBM 360/67 computer, using various optional combinations of subroutines, showed significant differences in accuracy and computing time, with the optimum subroutine combination depending on mesh size.

Martin, E. D.↗

NASA NTRS1989

Personal computer study of finite-difference methods for the transonic small disturbance equation

Calculation of unsteady flow phenomena requires careful attention to the numerical treatment of the governing partial differential equations. The personal computer provides a convenient and useful tool for the development of meshes, algorithms, and boundary conditions needed to provide time accurate solution of these equations. The one-dimensional equation considered provides a suitable model for the study of wave propagation in the equations of transonic small disturbance potential flow. Numerical results for effects of mesh size, extent, and stretching, time step size, and choice of far-field boundary conditions are presented. Analysis of the discretized model problem supports these numerical results. Guidelines for suitable mesh and time step choices are given.

Bland, Samuel R.↗

NASA NTRS1990

Personal computer study of finite-difference methods for the transonic small disturbance equation

Calculation of unsteady flow phenomena requires careful attention to the numerical treatment to the governing partial differential equations. The personal computer provides a convenient and useful tool for the development of meshes, algorithms, and boundary conditions needed to provide time accurate solution of these equations. The one-dimensional equation considered provides a suitable model for the study of wave propagation in the equations of transonic small disturbance potential flow. Numerical results for effects of mesh size, extent, and stretching, time step size, and choice of far-field boundary conditions are presented. Analysis of the discretized model problem supports these numerical results. Guidelines for suitable mesh and time step choices are given.

Bland, Samuel R.↗

NASA NTRS1992

Examples Of Synthesis Of Dual-Shaped Reflectors

Report presents examples to demonstrate validity and utility of method of synthesis of offset dual-shape reflectors. Method of synthesis described by the authors in previous journal article. Current report reviews derivation of partial differential equations and iterative method of numerical solution. Discusses significance of starting point of numerical integration on each reflector surface; this point could be at center, on outer rim, or at interior point. Emphasizes that one of notable attributes of partial differential equations is speed with which they can be solved.

Galindo, Victor↗

NASA NTRS1985

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system.

Saltz, J. H.↗

NASA NTRS1988

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Saltz, Joel H.↗

DOE OSTI2022

Volumetric Rendering on Wavelet-Based Adaptive Grid

Numerical modeling of physical phenomena frequently involves processes across a wide range of spatial and temporal scales. In the last two decades, the advancements in wavelet-based numerical methodologies to solve partial differential equations, combined with the unique properties of wavelet analysis to resolve localized structures of the solution on dynamically adaptive computational meshes, make it feasible to perform large-scale numerical simulations of a variety of physical systems on a dynamically adaptive computational mesh that changes both in space and time. Volumetric visualization of the solution is an essential part of scientific computing, yet the existing volumetric visualization techniques do not take full advantage of multi-resolution wavelet analysis and are not fully tailored for visualization of a compressed solution on the wavelet-based adaptive computational mesh. Our objective is to explore the alternatives for the visualization of time-dependent data on space-time varying adaptive mesh using volume rendering while capitalizing on the available sparse data representation. Two alternative formulations are explored. The first one is based on volumetric ray casting of multi-scale datasets in wavelet space. Rather than working with the wavelets at the finest possible resolution, a partial inverse wavelet transform is performed as a preprocessing step to obtain scaling functions on a uniform grid at a user-prescribed resolution. As a result, a solution in physical space is represented by a superposition of scaling functions on a coarse regular grid and wavelets on an adaptive mesh. An efficient and accurate ray casting algorithm is based just on these coarse scaling functions. Additional details are added during the ray tracing by taking an appropriate number of wavelets into account based on support overlap with the interpolation point, wavelet coefficient magnitude, and other characteristics, such as opacity accumulation (front to back ordering) and deviation from frontal viewing direction. The second approach is based on complementing of wavelet-based adaptive mesh to the traditional Adaptive Mesh Refinement (AMR) mesh. Both algorithms are illustrated and compared to the existing volume visualization software for Rayleigh-Benard thermal convection and electron density data sets in terms of rendering time and visual quality for different data compression of both wavelet-based and AMR adaptive meshes.

Vezolainen, Alexei V.↗

NASA NTRS1989

A method for solution of the Euler-Bernoulli beam equation in flexible-link robotic systems

An efficient numerical method for solving the partial differential equation (PDE) governing the flexible manipulator control dynamics is presented. A finite-dimensional model of the equation is obtained through discretization in both time and space coordinates by using finite-difference approximations to the PDE. An expert program written in the Macsyma symbolic language is utilized in order to embed the boundary conditions into the program, accounting for a mass carried at the tip of the manipulator. The advantages of the proposed algorithm are many, including the ability to (1) include any distributed actuation term in the partial differential equation, (2) provide distributed sensing of the beam displacement, (3) easily modify the boundary conditions through an expert program, and (4) modify the structure for running under a multiprocessor environment.

Tzes, Anthony P.↗

NASA NTRS1978

Two-dimensional computer code for design assessment of SRM aft closure environment

A general interpolants method for constructing numerical analogs of the partial differential equations of continuum mechanics and which combines the best features of both finite element and finite difference methods was used in a two dimensional axisymmetric analysis of the space shuttle solid motor aft closure region flowfield. The numerical technique used is discussed as well as the results of the analysis. A steady state solution for the submerged nozzle nose region indicates the development of an area gas flow recirculation near the propellant burning surface boundary and in the region of the underside of the nozzle nose. This recirculation region remains spatially fixed for successive iterations (greater than 800) of the solution.

Freeman, J. A.↗

NASA NTRS1993

A multi-level solution algorithm for steady-state Markov chains

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing significant reductions in computation time, often an order of magnitude or more, relative to the Gauss-Seidel and optimal SOR algorithms for a variety of test problems. The multi-level method is compared and contrasted with the iterative aggregation-disaggregation algorithm of Takahashi.

Horton, Graham↗