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NASA NTRS1988

General methods for determining the linear stability of coronal magnetic fields

A time integration of a linearized plasma equation of motion has been performed to calculate the ideal linear stability of arbitrary three-dimensional magnetic fields. The convergence rates of the explicit and implicit power methods employed are speeded up by using sequences of cyclic shifts. Growth rates are obtained for Gold-Hoyle force-free equilibria, and the corkscrew-kink instability is found to be very weak.

Craig, I. J. D.↗

NASA NTRS1988

Backward differentiation approximations of nonlinear differential/algebraic systems

Finite-difference approximations of dynamical systems modeled by nonlinear, semiexplicit, differential/algebraic equations are analyzed. Convergence for the backward differentiation method is proved for index two and index three problems when the numerical initial values obey certain constraints. The appropriate asymptotic convergence rates and the leading error terms are determined.

Brenan, Kathryn E.↗

NASA NTRS1989

An investigation of cell centered and cell vertex multigrid schemes for the Navier-Stokes equations

Two efficient and robust finite-volume multigrid schemes for solving the Navier-Stokes equations are investigated. These schemes employ either a cell centered or a cell vertex discretization technique. An explicit Runge-Kutta algorithm is used to advance the solution in time. Acceleration techniques are applied to obtain faster steady-state convergence. Accuracy and convergence of the schemes are examined. Computational results for transonic airfoil flows are essentially the same, even for a coarse mesh. Both schemes exhibit good convergence rates for a broad range of artificial dissipation coefficients.

Radespiel, R.↗

NASA NTRS1988

High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows

The development of robust, accurate, and efficient implicit shock-capturing schemes for multidimensional compressible Navier-Stokes equations in the hypersonic and real gas flow regimes is presently undertaken by extending a class of implicit total variation-diminishing (TVD) schemes suitable for transonic and supersonic, multidimensional Euler and Navier-Stokes equations to hypersonic computations. Numerical aspects of TVD schemes are identified which affect the convergence rate for hypersonic Mach numbers and real gas flows, but which have a negligible effect on low Mach number or perfect gas flows.

Yee, H. C.↗

NASA NTRS1989

Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.

Keeling, Stephen L.↗

NASA NTRS1989

Design of optimally smoothing multi-stage schemes for the Euler equations

In this paper, a method is developed for designing multi-stage schemes that give optimal damping of high-frequencies for a given spatial-differencing operator. The objective of the method is to design schemes that combine well with multi-grid acceleration. The schemes are tested on a nonlinear scalar equation, and compared to Runge-Kutta schemes with the maximum stable time-step. The optimally smoothing schemes perform better than the Runge-Kutta schemes, even on a single grid. The analysis is extended to the Euler equations in one space-dimension by use of 'characteristic time-stepping', which preconditions the equations, removing stiffness due to variations among characteristic speeds. Convergence rates independent of the number of cells in the finest grid are achieved for transonic flow with and without a shock. Characteristic time-stepping is shown to be preferable to local time-stepping, although use of the optimally damping schemes appears to enhance the performance of local time-stepping. The extension of the analysis to the two-dimensional Euler equations is hampered by the lack of a model for characteristic time-stepping in two dimensions. Some results for local time-stepping are presented.

Van Leer, Bram↗

NASA NTRS1989

Solution of the incompressible Navier-Stokes equations using artificial compressibility methods

A modified artificial compressibility formulation is derived from a preconditioned low Mach number compressible formulation, and these two methods are compared on a two-dimensional laminar leading edge flow using a LBI/ADI solution algorithm. The two methods are essentially equivalent with appropriate preconditioning parameters and have the same convergence rates and efficiency, giving 4 orders of residual reduction in about 75 iterations with a vectorized CRAY-XMP run time of 20 seconds for 3000 grid points.

Briley, W. R.↗

NASA NTRS1989

Quotient-difference type generalizations of the power method and their analysis

The recursion relations that were proposed by W. F. Ford and A. Sidi (Appl. Numer. Math, 4 (1988), pp. 477-489) for implementing vector extrapolation methods are used for devising generalizations of the power method for linear operators. These generalizations are shown to produce approximations to largest eigenvalues of a linear operator under certain conditions. They are similar in form to the quotient-difference algorithm and share similar convergence properties with the latter. These convergence properties also resemble those obtained for the basic LR and QR algorithms. Finally, it is shown that the convergence rate produced by one fo these generalizations is twice as fast for normal operators as it is for nonnormal operators.

Sidi, Avram↗

NASA NTRS1989

Extended Kalman filter for attitude estimation of the earth radiation budget satellite

The design and testing of an Extended Kalman Filter (EKF) for ground attitude determination, misalignment estimation and sensor calibration of the Earth Radiation Budget Satellite (ERBS) are described. Attitude is represented by the quaternion of rotation and the attitude estimation error is defined as an additive error. Quaternion normalization is used for increasing the convergence rate and for minimizing the need for filter tuning. The development of the filter dynamic model, the gyro error model and the measurement models of the Sun sensors, the IR horizon scanner and the magnetometers which are used to generate vector measurements are also presented. The filter is applied to real data transmitted by ERBS sensors. Results are presented and analyzed and the EKF advantages as well as sensitivities are discussed. On the whole the filter meets the expected synergism, accuracy and robustness.

Deutschmann, Julie↗

NASA NTRS1990

An upwind-biased, point-implicit relaxation algorithm for viscous, compressible perfect-gas flows

An upwind-biased, point-implicit relaxation algorithm for obtaining the numerical solution to the governing equations for three-dimensional, viscous, compressible, perfect-gas flows is described. The algorithm is derived using a finite-volume formulation in which the inviscid components of flux across cell walls are described with Roe's averaging and Harten's entropy fix with second-order corrections based on Yee's Symmetric Total Variation Diminishing scheme. Viscous terms are discretized using central differences. The relaxation strategy is well suited for computers employing either vector or parallel architectures. It is also well suited to the numerical solution of the governing equations on unstructured grids. Because of the point-implicit relaxation strategy, the algorithm remains stable at large Courant numbers without the necessity of solving large, block tri-diagonal systems. Convergence rates and grid refinement studies are conducted for Mach 5 flow through an inlet with a 10 deg compression ramp and Mach 14 flow over a 15 deg ramp. Predictions for pressure distributions, surface heating, and aerodynamics coefficients compare well with experiment data for Mach 10 flow over a blunt body.

Gnoffo, Peter A.↗

NASA NTRS1990

Solution algorithms for the two-dimensional Euler equations on unstructured meshes

The objective of the study was to analyze implicit techniques employed in structured grid algorithms for solving two-dimensional Euler equations and extend them to unstructured solvers in order to accelerate convergence rates. A comparison is made between nine different algorithms for both first-order and second-order accurate solutions. Higher-order accuracy is achieved by using multidimensional monotone linear reconstruction procedures. The discussion is illustrated by results for flow over a transonic circular arc.

Whitaker, D. L.↗

NASA NTRS1990

Adaptive remeshing method for finite-element thermal analysis

A finite-element remeshing approach that makes use of quadrilateral and triangular elements is described. The approach uses the solution on a previous mesh to create a new mesh. Meshes are completely unstructured with highly refined elements in regions of steep gradients and larger elements where gradients are smaller. Studies of convergence rates for heat conduction problems with exact solutions show that for problems with highly localized solution variations, the remeshing approach gives smaller solution errors with fewer unknowns than refinement of uniform, structured meshes.

Thornton, Earl A.↗

NASA NTRS1990

High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows

Hypersonic computations are presently conducted with an extension of a class of high-resolution implicit TVD algorithms suited to transonic multidimensional Euler and Navier-Stokes equations. These conservative shock-capturing schemes, which are spatially second- and third-order, may be first- and second-order accurate in time and suitable for either steady or unsteady calculations. Attention is given to the enhancement of hypersonic flows' convergence rate and stability; accuracy and efficiency is achieved by these means for very complex two-dimensional hypersonic viscous and inviscid shock interactions.

Yee, H. C.↗

NASA NTRS1990

Enthalpy damping for high Mach number Euler solutions

An improvement on the enthalpy damping procedure currently in use in solving supersonic flow fields is described. A correction based on entropy values is shown to produce a very efficient scheme for simulation of high Mach number three-dimensional flows. Substantial improvements in convergence rates have been achieved by incorporating this enthalpy damping scheme in a finite-volume Runge-Kutta method for solving the Euler equations. Results obtained for blended wing-body geometries at very high Mach numbers are presented.

Moitra, Anutosh↗

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 NTRS1990

Numerical analysis of secondary flow in a two-stage turbine

The three-dimensional viscous average passage flow in the Pratt and Whitney alternate design Space Shuttle Main Engine fuel turbine has been simulated. The effect of secondary flows generated by upstream blade rows on the performance of downstream blade rows is studied. The numerical results are compared to the design intent to validate improved models in the average passage equations. Analysis of the results centers on the primary spanwise mixing mechanism in this low aspect ratio turbine. A multigrid method has also been incorporated to improve the overall convergence rate of the numerical algorithm.

Kirtley, K. R.↗

NASA NTRS1990

Factorization and the synthesis of optimal feedback gains for distributed parameter systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Milman, Mark H.↗

NASA NTRS1990

Regularized Chapman-Enskog expansion for scalar conservation laws

Rosenau has recently proposed a regularized version of the Chapman-Enskog expansion of hydrodynamics. This regularized expansion resembles the usual Navier-Stokes viscosity terms at law wave-numbers, but unlike the latter, it has the advantage of being a bounded macroscopic approximation to the linearized collision operator. The behavior of Rosenau regularization of the Chapman-Enskog expansion (RCE) is studied in the context of scalar conservation laws. It is shown that thie RCE model retains the essential properties of the usual viscosity approximation, e.g., existence of traveling waves, monotonicity, upper-Lipschitz continuity..., and at the same time, it sharpens the standard viscous shock layers. It is proved that the regularized RCE approximation converges to the underlying inviscid entropy solution as its mean-free-path epsilon approaches 0, and the convergence rate is estimated.

Schochet, Steven↗