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All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Advances in upwind relaxation methods

Numerical techniques for solving the compressible Euler and Navier-Stokes equations are discussed with an emphasis on characteristic-based schemes. Two popular approaches, flux difference splitting and flux vector splitting, are described in one-dimensional Cartesian coordinates and then extended to three-dimensional generalized coordinates. A technique for increasing the spatial accuracy is presented, followed by a discussion of numerical dissipation mechanisms. An introduction to the use of implicit time integration schemes for accelerating the convergence rate to steady-state solutions including Newton's method, relaxation strategies, and approximate factorization techniques and their implementation on a vector processor concludes the chapter.

Walters, R. W.↗
NASA NTRS1984

Relaxation methods for unfactored implicit upwind schemes

Relaxation methods are presented for unfactored implicit upwind schemes for hyperbolic equations. The theoretical bases are explained using linear and nonlinear scalar equations; construction of the method for the unsteady Euler equations (nonlinear system) is but a natural extension. One of the important advantages of the above methods vis a vis factored implicit schemes is the possibility of faster convergence to steady state, as illustrated by the results. Several classes of relaxation schemes such as pointwise, linewise, Gauss-Seidel, and non-Gauss-Seidel methods are discussed, along with various strategies for convergence.

Chakravarthy, S. R.↗
NASA NTRS1987

Determination of diffusion coefficients in polypyrrole thin films using a current pulse relaxation method

The current pulse E sub oc relaxation method and its application to the determination of diffusion coefficients in electrochemically synthesized polypyrrole thin films is described. Diffusion coefficients for such films in Et4NBF4 and MeCN are determined for a series of submicron film thicknesses. Measurement of the double-layer capacitance, C sub dl, and the resistance, R sub u, of polypyrrole thin films as a function of potential obtained with the galvanostatic pulse method is reported. Measurements of the electrolyte concentration in reduced polypyrrole films are also presented to aid in the interpretation of the data.

Penner, Reginald M.↗
NASA NTRS1972

A general relaxation method for inverse solution of the full radiative transfer equation.

The relaxation method for the inverse solution of the full radiative transfer equation is generalized to solve for all atmospheric parameters which appear in the integrand as functions or functionals, without any a priori information related to the expected solution. The method is illustrated by examples in the earth's atmosphere for the determination of water vapor mixing ratio profiles from observations in the 6.3-micron band.

Chahine, M. T.↗
NASA NTRS1973

Relaxation methods for transonic flow about wing-cylinder combinations and lifting swept wings

The mixed elliptic-hyperbolic relaxation method for obtaining steady-state solutions to two-dimensional transonic potential equations is extended to the transonic small disturbance equation in three dimensions. In particular, transonic flow is considered both about nonlifting wing-cylinder combinations and over thin lifting wings with sweep and taper. The treatment is restricted to freestream Mach numbers less than one and to wings with subsonic trailing edges.

Bailey, F. R.↗
NASA NTRS1981

A preliminary study of relaxation methods for the inviscid conservative gasdynamics equations using flux splitting

Plus minus flux vector split schemes are combined with the multigrid relaxation method to obtain fast rates of iterative (or steady state) convergence. On model nonlifting biconvex airfoil flow, steady state convergence is obtained using multigrid in 40 iteractions for subsonic cases and in 120 iterations for transonic cases. A 65 x 33 stretched grid is used. However, the multigrid scheme used here (and there are numerous variations) sometimes could not be made to work for transonic cases with a large number of supersonic points. Moreover, the multigrid method is only several times faster than the conventional algorithm which itself could perhaps be accelerated by other means. Nethertheless, with proper adjustment the multigrid method will ultimately lead to very fast rates of iterative convergence.

Steger, J. L.↗
NASA NTRS1975

Relaxation methods in fluid mechanics

The present work considers the iterative solution of a coupled set of difference equations and examines methods that carry successive approximates to a state that is invariant with further iteration and independent of the initial guess. Methods are studied with regard to their efficiency and economy of computer resources. The basic principles of classical relaxation are set forth, with attention confined to linear elliptic equations. This discussion involves the evaluation of the spectral radius that is the magnitude of the eigenvalue with largest modulus. The subject of relaxation is then related to the study of ordinary differential equations and hyperbolic partial differential equations. Problems that occur when linearly dependent eigenvectors appear in the relaxation matrix are discussed, leading to multiply connected eigenvalues in the Jordan canonical form. Finally, a brief survey of relaxation methods used in aerodynamics is given.

Lomax, H.↗
NASA NTRS1952

A biharmonic relaxation method for calculating thermal stress in cooled irregular cylinders

A numerical method was developed for calculating thermal stresses in irregular cylinders cooled by one or more internal passages. The use of relaxation methods and elementary methods of finite differences was found to give approximations to the correct values when compared with previously known solutions for concentric circular cylinders possessing symmetrical and asymmetrical temperature distributions.

Holms, Arthur G↗
NASA NTRS1994

Data-parallel lower-upper relaxation method for reacting flows

The implicit lower-upper symmetric Gauss-Seidel (LU-SGS) method of Yoon and Jameson is modified for use on massively parallel computers. The method has been implemented on the Thinking Machines CM-5 and the MasPar MP-1 and MP-2, where large percentages of the theoretical peak floating point performance are obtained. It is shown that the new data-parallel LU relaxation method has better convergence properties than the original method for two different inviscid compressible flow simulations. The convergence is also improved for five-species reacting air computations. The performance of the method on various partitions of the CM-5 and on the MasPar computers is discussed. The new method shows promise for the efficient simulation of very large perfect gas and reacting flows.

Candler, Graham V.↗
NASA NTRS1989

Upwind relaxation methods for the Navier-Stokes equations using inner iterations

A supersonic and subsonic test problem is solved using an upwind line relaxation algorithm for the Navier-Stokes equations which employs inner iterations. In the supersonic test case, the inner iteration method was found to require only about 65 percent of the CPU time which was required by the most efficient line relaxation method without inner iterations. In the subsonic test case, poor matrix conditioning resulted in less efficiency than line relaxation methods employing a more conventional CPU-saving strategy.

Taylor, Arthur C., III↗
NASA NTRS1992

Upwind relaxation methods for the Navier-Stokes equations using inner iterations

A subsonic and a supersonic problem are respectively treated by an upwind line-relaxation algorithm for the Navier-Stokes equations using inner iterations to accelerate steady-state solution convergence and thereby minimize CPU time. While the ability of the inner iterative procedure to mimic the quadratic convergence of the direct solver method is attested to in both test problems, some of the nonquadratic inner iterative results are noted to have been more efficient than the quadratic. In the more successful, supersonic test case, inner iteration required only about 65 percent of the line-relaxation method-entailed CPU time.

Taylor, Arthur C., III↗
NASA NTRS1975

On the computation of two- and three-dimensional steady transonic flows by relaxation methods

The paper is concerned with the application of the Murman and Cole (1971) relaxation scheme to steady, inviscid transonic flow problems in two and three dimensions. This scheme, which automatically accounts for weak shock waves, uses separate difference operators in elliptic and hyperbolic regions. The details of the scheme are described in terms of the original small disturbance formulation of Murman and Cole. In particular, Murman's recent (1973) introduction of fully conservative difference operators to obtain the correct shock jumps is examined. The extension to treating the exact isentropic equation is then covered with special attention given to Jameson's (to appear) rotated difference scheme for supersonic flow regions. The bulk of the discussion is related to two-dimensional procedures, and some comparisons with experiment are made, with emphasis on the effects of viscosity and wind-tunnel walls. Application of the Murman-Cole scheme is then discussed for small disturbances in three dimensions.

Bailey, F. R.↗
NASA NTRS1998

Real Gas Computation Using an Energy Relaxation Method and High-Order WENO Schemes

In this paper, we use a recently developed energy relaxation theory by Coquel and Perthame and high order weighted essentially non-oscillatory (WENO) schemes to simulate the Euler equations of real gas. The main idea is an energy decomposition into two parts: one part is associated with a simpler pressure law and the other part (the nonlinear deviation) is convected with the flow. A relaxation process is performed for each time step to ensure that the original pressure law is satisfied. The necessary characteristic decomposition for the high order WENO schemes is performed on the characteristic fields based on the first part. The algorithm only calls for the original pressure law once per grid point per time step, without the need to compute its derivatives or any Riemann solvers. Both one and two dimensional numerical examples are shown to illustrate the effectiveness of this approach.

Montarnal, Philippe↗
NASA NTRS1987

Relaxation method of compensation in an optical correlator

An iterative method is proposed for the sharpening of programmable filters in a 4-f optical correlator. Continuously variable spatial light modulators (SLMs) permit the fine adjustment of optical processing filters so as to compensate for the departures from ideal behavior of a real optical system. Although motivated by the development of continuously variable phase-only SLMs, the proposed sharpening method is also applicable to amplitude modulators and, with appropriate adjustments, to binary modulators as well. A computer simulation is presented that illustrates the potential effectiveness of the method: an image is placed on the input to the correlator, and its corresponding phase-only filter is adjusted (allowed to relax) so as to produce a progressively brighter and more centralized peak in the correlation plane. The technique is highly robust against the form of the system's departure from ideal behavior.

Juday, Richard D.↗
NASA NTRS1993

Line relaxation methods for the solution of 2D and 3D compressible flows

An implicit finite element based algorithm for the compressible Navier-Stokes equations is outlined, and the solution of the resulting equation by a line relaxation on general meshes of triangles or tetrahedra is described. The problem of generating and adapting unstructured meshes for viscous flows is reexamined, and an approach for both 2D and 3D simulations is proposed. An efficient approach appears to be the use of an implicit/explicit procedure, with the implicit treatment being restricted to those regions of the mesh where viscous effects are known to be dominant. Numerical examples demonstrating the computational performance of the proposed techniques are given.

Hassan, O.↗