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NASA NTRS1975

Special Perturbations Using Back-Correction Methods of Numerical Integration

A new class of linear multistep methods for numerical integration of differential equations is reported that permits satellite computation solutions to be corrected at certain points in the past as the integration advances in time. Algorithms have been developed for the solution of both first- and second-order differential equations. The back correction method appears to be more efficient than classical methods when dominant and perturbing forces can be separated.

Feagin, T.↗
NASA NTRS1978

An extension of A-stability to alternating direction implicit methods

An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented.

Warming, R. F.↗
NASA NTRS1973

Analysis and optimization of cyclic methods in orbit computation

The mathematical analysis and computation of the K=3, order 4; K=4, order 6; and K=5, order 7 cyclic methods and the K=5, order 6 Cowell method and some results of optimizing the 3 backpoint cyclic multistep methods for solving ordinary differential equations are presented. Cyclic methods have the advantage over traditional methods of having higher order for a given number of backpoints while at the same time having more free parameters. After considering several error sources the primary source for the cyclic methods has been isolated. The free parameters for three backpoint methods were used to minimize the effects of some of these error sources. They now yield more accuracy with the same computing time as Cowell's method on selected problems. This work is being extended to the five backpoint methods. The analysis and optimization are more difficult here since the matrices are larger and the dimension of the optimizing space is larger. Indications are that the primary error source can be reduced. This will still leave several parameters free to minimize other sources.

Pierce, S.↗
NASA NTRS2018

Solution of Ordinary Differential Equations in Gradient-Based Multidisciplinary Design Optimization

A gradient-based approach to multidisciplinary design optimization enables efficient scalability to large numbers of design variables. However, the need for derivatives causes difficulties when integrating ordinary differential equations (ODEs) in models. To simplify this, we propose the use of the general linear methods framework, which unifies all Runge-Kutta and linear multistep methods. This approach enables rapid implementation of integration methods without the need to differentiate each one, even in a gradient-based optimization context. We also develop a new parallel time integration algorithm that enables vectorization across time steps. We present a set of benchmarking results using a stiff ODE, a non-stiff nonlinear ODE, and an orbital dynamics ODE, and compare integration methods. In a modular gradient-based multidisciplinary design optimization context, we find that the new parallel time integration algorithm with high-order implicit methods, especially Gauss-Legendre collocation, is the best choice for a broad range of problems.

Hwang, John T.↗
NASA NTRS1976

Conservation properties of numerical integration methods for systems of ordinary differential equations

If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.

Rosenbaum, J. S.↗
NASA NTRS1995

The construction of high-accuracy schemes for acoustic equations

An accuracy analysis of various high order schemes is performed from an interpolation point of view. The analysis indicates that classical high order finite difference schemes, which use polynomial interpolation, hold high accuracy only at nodes and are therefore not suitable for time-dependent problems. Thus, some schemes improve their numerical accuracy within grid cells by the near-minimax approximation method, but their practical significance is degraded by maintaining the same stencil as classical schemes. One-step methods in space discretization, which use piecewise polynomial interpolation and involve data at only two points, can generate a uniform accuracy over the whole grid cell and avoid spurious roots. As a result, they are more accurate and efficient than multistep methods. In particular, the Cubic-Interpolated Psuedoparticle (CIP) scheme is recommended for computational acoustics.

Tang, Lei↗
NASA NTRS1981

A modified Dodge algorithm for the parabolized Navier-Stokes equation and compressible duct flows

A revised version of Dodge's split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition. Qualitive agreement with analytical predictions and experimental results was obtained for some flows with well-known solutions.

Cooke, C. H.↗
NASA NTRS1983

A modified Dodge algorithm for the parabolized Navier-Stokes equations and compressible duct flows

A revised version of Dodge's split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition. Qualitative agreement with analytical predictions and experimental results was obtained for some flows with well-known solutions. Previously announced in STAR as N82-16363

Cooke, C. H.↗
NASA NTRS1981

A modified dodge algorithm for the parabolized Navier-Stokes equations and compressible duct flows

A revised version of a split-velocity method for numerical calculation of compressible duct flow was developed. The revision incorporates balancing of mass flow rates on each marching step in order to maintain front-to-back continuity during the calculation. The (checkerboard) zebra algorithm is applied to solution of the three-dimensional continuity equation in conservative form. A second-order A-stable linear multistep method is employed in effecting a marching solution of the parabolized momentum equations. A checkerboard successive overrelaxation iteration is used to solve the resulting implicit nonlinear systems of finite-difference equations which govern stepwise transition.

Cooke, C. H.↗
NASA NTRS1973

Algorithms for changing the step size

Approximately ten different ways for changing the step size used by multistep methods are enumerated, and their good and bad features are compared. More efficient algorithms are given for the difference formulations of a frequently used halving and doubling process, and a cure for the instability inherent in this halving process is proposed.

Krogh, F. T.↗
NASA NTRS1974

Changing stepsize in the integration of differential equations using modified divided differences

Multistep methods for solving differential equations based on numerical integration formulas or numerical differentiation formulas (for stiff equations) require special provision for changing the stepsize. New algorithms are given which make the use of modified divided differences an attractive way to carry out the change in stepsize for such methods. Error estimation and some of the important factors in stepsize selection and the selection of integration order are also considered.

Krogh, F. T.↗
NASA NTRS1976

Heat shield characterization: Outer planet atmospheric entry probe

A full scale carbon phenolic heat shield was fabricated for the Outer Planet Probe in order to demonstrate the feasibility of molding large carbon phenolic parts with a new fabrication processing method (multistep). The sphere-cone heat shield was molded as an integral unit with the nose cap plies configured into a double inverse chevron shape to achieve the desired ply orientation. The fabrication activity was successful and the feasibility of the multistep processing technology was established. Delaminations or unbonded plies were visible on the heat shield and resulted from excessive loss of resin and lack of sufficient pressure applied on the part during the curing cycle. A comprehensive heat shield characterization test program was conducted, including: nondestructive tests with the full scale heat shield and thermal and mechanical property tests with small test specimen.

Mezines, S. A.↗
NASA NTRS1981

Numerical approximation of boundary conditions with applications to inviscid equations of gas dynamics

A comprehensive overview of the state of the art of well-posedness and stability analysis of difference approximations for initial boundary value problems of the hyperbolic type is presented. The applicability of recent theoretical development to practical calculations for nonlinear gas dynamics is examined. The one dimensional inviscid gas dynamics equations in conservation law form are selected for numerical experiments. The class of implicit schemes developed from linear multistep methods in ordinary differential equations is chosen and the use of linear extrapolation as an explicit or implicit boundary scheme is emphasized. Specification of boundary data in the primitive variables and computation in terms of the conservative variables in the interior is discussed. Some numerical examples for the quasi-one-dimensional nozzle are given.

Yee, H. C.↗
NASA NTRS1981

Stable boundary approximations for a class of implicit schemes for the one-dimensional inviscid equations of gas dynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations for initial-boundary-value problems of the hyperbolic type. For the numerical experiments, select the one-dimensional inviscid gas-dynamic equations in conservation-law form is selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit boundary schemes is emphasized. Some numerical examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit versus implicit boundary schemes, unconditionally stable schemes, and underspecification or overspecification of boundary conditions.

Yee, H. C.↗
NASA NTRS1981

Recent advances in the development of implicit schemes for the equations of fluid dynamics

Innovations and extensions of implicit schemes for equations of fluid dynamics are presented. The notation and theory for linear multistep methods are reviewed, and extensions of work by Beam and Warming (1979) include the implementation of one-leg methods, ADI methods for equations with mixed derivatives, flux vector splitting, the P-dimensional wave equation, and boundary conditions. Numerical experiments indicate that implicit treatment of the boundary conditions is necessary for unconditional stability, and the improvement and implementation of the boundary condition theory should improve the implicit algorithms for gas dynamic equations.

Warming, R. F.↗
NASA NTRS1987

TDRSS OVLBI experiment - Data correlation and results

The use of an orbiting antenna in a VLBI experiment complicates the data correlation, due to the spacecraft motion and the presence of the satellite-to-ground link. For this experiment, using a TDRSS satellite, a multistep method was used to calculate delay and phase for correlation and fringe-fitting. Data from three sources: 1510-089, NRAO 530 (1730-130), and 174l-038 were successfully correlated. The maximum projected baseline achieved was 1.4 earth diameters, for NRAO 530, with baselines longer than one earth diameter obtained for all three sources. Fringe visibilities for the maximum projected baselines were: 0.15 (1510-089), 0.05 (NRAO 530), and 0.50 (1741-038). The coherence of baselines to the orbiting antenna was 84 percent for 700 seconds. For a dedicated orbiting VLBI observatory (e.g., QUASAT), excellent coherence can be expected for integration times of several hundred seconds and observing frequencies as high as 20-30 GHz.

Linfield, R. P.↗
NASA NTRS1990

Geometry control in adaptive truss structures

Forward and inverse kinematics equations are derived for the large geometry maneuver of adaptive trusses. A new algorithm based on higher-order multistep methods is proposed as a means of computing the length control for a described large geometry maneuver. The algorithm is shown to improve in computational speed at least five times over the algorithm presented in an earlier paper. The acceleration in control computation and other features such as varying velocity profiles and curved trajectories are illustrated by simulation results.

Ramesh, A. V.↗
NASA NTRS1994

Global Asymptotic Behavior of Iterative Implicit Schemes

The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing three models of 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed using the theory of dynamical systems. The iterative procedures include simple iteration and full and modified Newton iterations. The results are compared with standard Runge-Kutta explicit methods, a noniterative implicit procedure, and the Newton method of solving the steady part of the ODEs. Studies showed that aside from exhibiting spurious asymptotes, all of the four implicit LMMs can change the type and stability of the steady states of the differential equations (DEs). They also exhibit a drastic distortion but less shrinkage of the basin of attraction of the true solution than standard nonLMM explicit methods. The simple iteration procedure exhibits behavior which is similar to standard nonLMM explicit methods except that spurious steady-state numerical solutions cannot occur. The numerical basins of attraction of the noniterative implicit procedure mimic more closely the basins of attraction of the DEs and are more efficient than the three iterative implicit procedures for the four implicit LMMs. Contrary to popular belief, the initial data using the Newton method of solving the steady part of the DEs may not have to be close to the exact steady state for convergence. These results can be used as an explanation for possible causes and cures of slow convergence and nonconvergence of steady-state numerical solutions when using an implicit LMM time-dependent approach in computational fluid dynamics.

Yee, H. C.↗