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Results for “parabolic problems”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
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NASA NTRS2004

Error Analysis for Discontinuous Galerkin Method for Parabolic Problems

In the proposal, the following three objectives are stated: (1) A p-version of the discontinuous Galerkin method for a one dimensional parabolic problem will be established. It should be recalled that the h-version in space was used for the discontinuous Galerkin method. An a priori error estimate as well as a posteriori estimate of this p-finite element discontinuous Galerkin method will be given. (2) The parameter alpha that describes the behavior double vertical line u(sub t)(t) double vertical line 2 was computed exactly. This was made feasible because of the explicitly specified initial condition. For practical heat transfer problems, the initial condition may have to be approximated. Also, if the parabolic problem is proposed on a multi-dimensional region, the parameter alpha, for most cases, would be difficult to compute exactly even in the case that the initial condition is known exactly. The second objective of this proposed research is to establish a method to estimate this parameter. This will be done by computing two discontinuous Galerkin approximate solutions at two different time steps starting from the initial time and use them to derive alpha. (3) The third objective is to consider the heat transfer problem over a two dimensional thin plate. The technique developed by Vogelius and Babuska will be used to establish a discontinuous Galerkin method in which the p-element will be used for through thickness approximation. This h-p finite element approach, that results in a dimensional reduction method, was used for elliptic problems, but the application appears new for the parabolic problem. The dimension reduction method will be discussed together with the time discretization method.

Kaneko, Hideaki↗
NASA NTRS1985

Spectral methods in time for parabolic problems

A pseudospectral explicit scheme for solving linear, periodic, parabolic problems is described which has infinite accuracy both in time and in space. The high accuracy is achieved while the time resolution parameter M ( = ) (1/delta t) for time marching algorithm) and the space resolution parameter N B = O(1/detla x) have to satisfy M = O(N sup/+epsilon) epsilon O, compared to the common stability condition M = O(N sup 2) which has to be satisfied in any explicit finite order time algorithm.

Tal-Ezer, H.↗
NASA NTRS2004

Discontinuous Galerkin Finite Element Method for Parabolic Problems

In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of parallel ut(t) parallel Lz(omega) = parallel ut parallel2, for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.

Kaneko, Hideaki↗
NASA NTRS1986

An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in one space variable

The present adaptive FEM technique for convection-dominated problems is based on a Petrov-Galerkin scheme for spatial approximation, whose typical time-step employs test functions chosen to yield an approximate solution coinciding with the exact solutions at the finite element grid nodes. The derivation of truly local a posteriori error estimates is made possible by this procedure, which is also shown to be a very effective solver by the numerical examples presented.

Demkowicz, L.↗
NASA NTRS1988

Boundary identification for 2-D parabolic problems arising in thermal testing of materials

Problems on the identification of two-dimensional spatial domains arising in the detection and characterization of structural flaws in materials are considered. For a thermal diffusion system with external boundary input, observations of the temperature on the surface are used in an output least square approach. Parameter estimation techniques based on the method of mappings are discussed, and approximation schemes are developed based on a finite-element Galerkin approach. Theoretical convergence results for computational techniques are given, and the results are applied to the identification of two kinds of boundary shapes.

Banks, H. T.↗
NASA NTRS1991

Sinc-Galerkin estimation of diffusivity in parabolic problems

A fully Sinc-Galerkin method for the numerical recovery of spatially varying diffusion coefficients in linear partial differential equations is presented. Because the parameter recovery problems are inherently ill-posed, an output error criterion in conjunction with Tikhonov regularization is used to formulate them as infinite-dimensional minimization problems. The forward problems are discretized with a sinc basis in both the spatial and temporal domains thus yielding an approximate solution which displays an exponential convergence rate and is valid on the infinite time interval. The minimization problems are then solved via a quasi-Newton/trust region algorithm. The L-curve technique for determining an approximate value of the regularization parameter is briefly discussed, and numerical examples are given which show the applicability of the method both for problems with noise-free data as well as for those whose data contains white noise.

Smith, Ralph C.↗
NASA NTRS2004

A Discontinuous Galerkin Method for Parabolic Problems with Modified hp-Finite Element Approximation Technique

A recent paper is generalized to a case where the spatial region is taken in R(sup 3). The region is assumed to be a thin body, such as a panel on the wing or fuselage of an aerospace vehicle. The traditional h- as well as hp-finite element methods are applied to the surface defined in the x - y variables, while, through the thickness, the technique of the p-element is employed. Time and spatial discretization scheme based upon an assumption of certain weak singularity of double vertical line u(sub t) double vertical line 2, is used to derive an optimal a priori error estimate for the current method.

Kaneko, Hideaki↗
NASA NTRS1988

Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence

An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.

Rosen, I. G.↗
NASA NTRS1983

The linear regulator problem for parabolic systems

An approximation framework is presented for computation (in finite imensional spaces) of Riccati operators that can be guaranteed to converge to the Riccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.

Banks, H. T.↗
NASA NTRS1984

The linear regulator problem for parabolic systems

The paper presents an approximation framework for computation (in finite dimensional spaces) of Riccati operators that can be guaranteed to converge to the Ricccati operator in feedback controls for abstract evolution systems in a Hilbert space. It is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems.

Banks, H. T.↗
NASA NTRS1982

A penalty finite element algorithm for parabolic flow problems

The thin-layer simplification of the two-dimensional Navier-Stokes equations for steady viscous flow are developed using an order of magnitude analysis. A space marching finite element solution algorithm is developed, wherein the first order continuity effects are enforced as a penalty function differential constraint. Numerical results are presented to document accuracy and convergence features of the numerical solution algorithm.

Baker, A. J.↗
NASA NTRS1978

On acceleration of MacCormack's scheme

An acceleration of MacCormack's scheme due to Desideri and Tannehill is analyzed. It is found that for hyperbolic problems one cannot improve upon the efficiency of MacCormack's method. For parabolic problems the time step can be chosen arbitrarily large without loss of stability by an appropriate choice of the acceleration parameters. When applied to the heat equation this method is equivalent to both the Dufort-Frankel scheme and to MSOR.

Gottlieb, D.↗
NASA NTRS1973

Nonlinear problem of a shock-tube interaction-region boundary layer.

Description of a numerical method that dispenses with the recourse to linearization or 'momentum-integral-correction' schemes in solving nonlinear problems of boundary layer flow in the interaction region of a shock tube or of similar singular parabolic problems of shock-induced unsteady boundary layers. The accuracy of the numerical results obtainable is shown to be satisfactory.

Gupta, R. N.↗
NASA NTRS1976

Elliptic systems and numerical transformations

Properties of a transformation method, which was developed for solving fluid dynamic problems on general two dimensional regions, are discussed. These include construction error of the transformation and applications to mesh generation. An error and stability analysis for the numerical solution of a model parabolic problem is also presented.

Mastin, C. W.↗
NASA NTRS1996

Mixing and Transition Control Studied

Considerable progress in understanding nonlinear phenomena in both unbounded and wallbounded shear flow transition has been made through the use of a combination of high- Reynolds-number asymptotic and numerical methods. The objective of this continuing work is to fully understand the nonlinear dynamics so that ultimately (1) an effective means of mixing and transition control can be developed and (2) the source terms in the aeroacoustic noise problem can be modeled more accurately. Two important aspects of the work are that (1) the disturbances evolve from strictly linear instability waves on weakly nonparallel mean flows so that the proper upstream conditions are applied in the nonlinear or wave-interaction streamwise region and (2) the asymptotic formulations lead to parabolic problems so that the question of proper out-flow boundary conditions--still a research issue for direct numerical simulations of convectively unstable shear flows--does not arise. Composite expansion techniques are used to obtain solutions that account for both mean-flow-evolution and nonlinear effects. A previously derived theory for the amplitude evolution of a two-dimensional instability wave in an incompressible mixing layer (which is in quantitative agreement with available experimental data for the first nonlinear saturation stage for a plane-jet shear layer, a circular-jet shear layer, and a mixing layer behind a splitter plate) have been extended to include a wave-interaction stage with a three-dimensional subharmonic. The ultimate wave interaction effects can either give rise to explosive growth or an equilibrium solution, both of which are intimately associated with the nonlinear self-interaction of the three dimensional component. The extended theory is being evaluated numerically. In contrast to the mixing-layer situation, earlier comparisons of theoretical predictions based on asymptotic methods and experiments in wall-bounded shear-flow transition have been somewhat lacking in one aspect or another. The current work strongly suggests that the main weakness is the underlying asymptotic representation of the linear "part" of the problem and not the explicit modeling of the nonlinear/wave-interaction effects. Consequently, the long-wave-length/high-Reynolds-number asymptotic limit for the Blasius boundary-layer stability problem was reexamined, and a new dispersion relationship for the instability waves that is uniformly valid for both the upper- and lower branch regions to the required order of approximation was obtained. A comparison with numerical results, obtained by solving the Orr-Sommerfeld stability problem, shows that the asymptotic formula provides surprisingly good results, even for values of the frequency parameter usually encountered in experimental investigations. This is particularly evident in the dynamically important upper-branch region, where much of the nonlinear interactions in transition experiments are believed to take place. The result is important in that it can be used to greatly improve the accuracy of weakly nonlinear critical-layer-based theories, and a consistent nonlinear theory is currently under evaluation.

Source record↗
NASA NTRS1989

On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented.

Gallopoulos, E.↗