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NASA NTRS1979

Error analysis of finite difference schemes applied to hyperbolic initial boundary value problems

Finite difference methods for the numerical solution of mixed initial boundary value problems for hyperbolic equations are studied. The reported investigation has the objective to develop a technique for the total error analysis of a finite difference scheme, taking into account initial approximations, boundary conditions, and interior approximation. Attention is given to the Cauchy problem and the initial approximation, the homogeneous problem in an infinite strip with inhomogeneous boundary data, the reflection of errors in the boundaries, and two different boundary approximations for the leapfrog scheme with a fourth order accurate difference operator in space.

Skollermo, G.↗

NASA NTRS1979

Vector processor algorithms for transonic flow calculations

This paper discusses a number of algorithms for solving the transonic full-potential equation in conservative form on a vector computer, such as the CDC STAR-100 or the CRAY-1. Recent research with the 'artificial density' method for transonics has led to development of some new iteration schemes which take advantage of vector-computer architecture without suffering significant loss of convergence rate. Several of these more promising schemes are described and 2-D and 3-D results are shown comparing the computational rates on the STAR and CRAY vector computers, and the CYBER-175 serial computer. Schemes included are: (1) Checkerboard SOR, (2) Checkerboard Leapfrog, (3) odd-even vertical line SOR, and (4) odd-even horizontal line SOR.

South, J. C., Jr.↗

NASA NTRS1979

Explicit large time-step schemes for the shallow water equations

Modifications to explicit finite difference schemes for solving the shallow water equations for meteorological applications by increasing the time step for the fast gravity waves are analyzed. Terms associated with the gravity waves in the shallow water equations are treated on a coarser grid than those associated with the slow Rossby waves, which contain much more of the available energy and must be treated with higher accuracy, enabling a several-fold increase in time step without degrading the accuracy of the solution. The method is presented in Cartesian and spherical coordinates for a rotating earth, using generalized leapfrog, frozen coefficient, and Fourier filtering finite difference schemes. Computational results verify the numerical stability of the approach.

Turkel, E.↗

NASA NTRS1980

On time discretizations for spectral methods

New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.

Gottlieb, D.↗

NASA NTRS1981

Stability of pseudospectral and finite-difference methods for variable coefficient problems

It is shown that pseudospectral approximation to a special class of variable coefficient one-dimensional wave equations is stable and convergent even though the wave speed changes sign within the domain. Computer experiments indicate similar results are valid for more general problems. Similarly, computer results indicate that the leapfrog finite-difference scheme is stable even though the wave speed changes sign within the domain. However, both schemes can be asymptotically unstable in time when a fixed spatial mesh is used.

Gottlieb, D.↗

NASA NTRS1981

Direct numerical simulations of turbulent shear flows

Numerical simulations of wakes of axisymmetric bodies and of turbulent mixing layers are reported. The flows were assumed to be statistically homogeneous in the mean flow direction, in concert with experimental data and the self-similarity theorem. The nonlinear Navier-Stokes equations were solved by a pseudo-spectral numerical method using a 32 x 32 x 33 point grid and an algorithm for fast Fourier transforms and inverse transforms. Leapfrog time differencing was employed on nonlinear terms and time differencing on viscous terms. Towed wakes and wakes behind a self-propelled body were simulated, showing that the towed wakes exhibited a proper temporal behavior after an initial period of adjustment, including the development of a kurtosis near the wake edge, which is experimentally verifiable. The mixing-layer simulation displayed the laboratory demonstrated presence of large scale features such as vortex cores, while the lateral coherence was weak.

Metcalfe, R. W.↗

NASA NTRS1983

The effects of the Asselin time filter on numerical solutions to the linearized shallow-water wave equations

In the present investigation, a one-dimensional linearized analysis is used to determine the effect of Asselin's (1972) time filter on both the computational stability and phase error of numerical solutions for the shallow water wave equations, in cases with diffusion but without rotation. An attempt has been made to establish the approximate optimal values of the filtering parameter nu for each of the 'lagged', Dufort-Frankel, and Crank-Nicholson diffusion schemes, suppressing the computational wave mode without materially altering the physical wave mode. It is determined that in the presence of diffusion, the optimum filter length depends on whether waves are undergoing significant propagation. When moderate propagation is present, with or without diffusion, the Asselin filter has little effect on the spatial phase lag of the physical mode for the leapfrog advection scheme of the three diffusion schemes considered.

Schlesinger, R. E.↗

NASA NTRS1984

Memory efficient solution of the primitive equations for numerical weather prediction on the CYBER 205

Numerical Weather Prediction (NWP), for both operational and research purposes, requires only fast computational speed but also large memory. A technique for solving the Primitive Equations for atmospheric motion on the CYBER 205, as implemented in the Mesoscale Atmospheric Simulation System, which is fully vectorized and requires substantially less memory than other techniques such as the Leapfrog or Adams-Bashforth Schemes is discussed. The technique presented uses the Euler-Backard time marching scheme. Also discussed are several techniques for reducing computational time of the model by replacing slow intrinsic routines by faster algorithms which use only hardware vector instructions.

Tuccillo, J. J.↗

NASA NTRS1985

Stability of interfaces with mesh refinement

A study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite-difference approximations. First, a root condition which implies stability for the initial-boundary value problem for this type of interface is derived. From the root condition, the stability of several interface equations is proved, using the maximum principle. In some cases, the final verification steps can be done analytically; in other cases, a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle. Using this method, stability for Lax-Wendroff with all the interface conditions considered, and for Leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved.

Berger, M. J.↗

NASA NTRS1985

Electromagnetic particle simulation codes

Electromagnetic particle simulations solve the full set of Maxwell's equations. They thus include the effects of self-consistent electric and magnetic fields, magnetic induction, and electromagnetic radiation. The algorithms for an electromagnetic code which works directly with the electric and magnetic fields are described. The fields and current are separated into transverse and longitudinal components. The transverse E and B fields are integrated in time using a leapfrog scheme applied to the Fourier components. The particle pushing is performed via the relativistic Lorentz force equation for the particle momentum. As an example, simulation results are presented for the electron cyclotron maser instability which illustrate the importance of relativistic effects on the wave-particle resonance condition and on wave dispersion.

Pritchett, P. L.↗

NASA NTRS1988

Navier-Stokes simulations of axisymmetric vortex rings

The solution of the incompressible Navier-Stokes equations by means of a spectral method is used in the present numerical investigation of viscous axisymmetric vortex rings, in the cases of a single ring over the 0.001-1000 Re number range, and a pair of interacting rings. The propagation speeds of vortex rings of varying Re are computed and compared with the results of Saffman's (1970) theory; the present error estimate is found to be smaller than Saffman's, and to decrease with increasing Re at fixed core/ring radius ratio. The 'leapfrogging' of two vortex rings of equal sign and Re=1000 each is observed, indicating severe strain of the vortex core through the first pass and merging during the second pass.

Stanaway, S. K.↗

NASA NTRS1988

A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass.

Stanaway, S. K.↗

NASA NTRS1989

Dynamics of a class of vortex rings

The contour dynamics method is extended to vortex rings with vorticity varying linearly from the symmetry axis. An elliptic core model is also developed to explain some of the basic physics. Passage and collisions of two identical rings are studied focusing on core deformation, sound generation and stirring of fluid elements. With respect to core deformation, not only the strain rate but how rapidly it varies is important and accounts for greater susceptibility to vortex tearing than in two dimensions. For slow strain, as a passage interaction is completed and the strain relaxes, the cores return to their original shape while permanent deformations remain for rapidly varying strain. For collisions, if the strain changes slowly the core shapes migrate through a known family of two-dimensional steady vortex pairs up to the limiting member of the family. Thereafter energy conservation does not allow the cores to maintain a constant shape. For rapidly varying strain, core deformation is severe and a head-tail structure in good agreement with experiments is formed. With respect to sound generation, good agreement with the measured acoustic signal for colliding rings is obtained and a feature previously thought to be due to viscous effects is shown to be an effect of inviscid core deformation alone. For passage interactions, a component of high frequency is present. Evidence for the importance of this noise source in jet noise spectra is provided. Finally, processes of fluid engulfment and rejection for an unsteady vortex ring are studied using the stable and unstable manifolds. The unstable manifold shows excellent agreement with flow visualization experiments for leapfrogging rings suggesting that it may be a good tool for numerical flow visualization in other time periodic flows.

Shariff, Karim↗

NASA NTRS1991

Spectral Method For Simulation Of Vortex Rings

Method of computation relying on spectral basis functions developed especially for simulation of axisymmetric vortex rings in incompressible, viscous fluid with quiescent far field. Contributes to understanding of flows in and around vortex rings during long propagation times, including such theoretically and practically important phenomena as drift and expansion of ring, "leapfrogging" and coalescence of two rings, and shedding of vorticity into wake of propagating ring.

Stanaway, S. K.↗

NASA NTRS1992

Vortex rings

The vortex-ring problem in fluid mechanics is examined generally in terms of formation, the steady state, the duration of the rings, and vortex interactions. The formation is studied by examining the generation of laminar and turbulent vortex rings and their resulting structures with attention given to the three stages of laminar ring development. Inviscid dynamics is addressed to show how core dynamics affects overall ring motion, and laminar vortex structures are described in two dimensions. Viscous and inviscid structures are related in terms of 'leapfrogging', head-on collisions, and collisions with a no-slip wall. Linear instability theory is shown to successfully describe observational data, although late stages in the breakdown are not completely understood. This study of vortex rings has important implications for key aerodynamic issues including sound generation, transport and mixing, and vortex interactions.

Shariff, Karim↗

NASA NTRS1992

A comparison of the impact of two time-differencing schemes on the NASA-GLAS climate model

Evidence is presented of the sensitivity of a GCM to the time-differencing scheme employed when the physical parameterizations and space discretization are not changed. Two-time marching schemes, the leapfrog and the Matsuno schemes, are analyzed and tested on the NASA-Goddard Laboratory for Atmospheric Studies fourth-order GCM in terms of the behavior and stability of two-month-averaged fields. Linear analysis indicates that Rossby waves are slightly accelerated and slightly damped when the Matsuno scheme is utilized and that these effects are scale selective, being smallest for the longest waves.

Pfeffer, Richard L.↗

NASA NTRS1992

A domain-specific design architecture for composite material design and aircraft part redesign

Advanced composites have been targeted as a 'leapfrog' technology that would provide a unique global competitive position for U.S. industry. Composites are unique in the requirements for an integrated approach to designing, manufacturing, and marketing of products developed utilizing the new materials of construction. Numerous studies extending across the entire economic spectrum of the United States from aerospace to military to durable goods have identified composites as a 'key' technology. In general there have been two approaches to composite construction: build models of a given composite materials, then determine characteristics of the material via numerical simulation and empirical testing; and experience-directed construction of fabrication plans for building composites with given properties. The first route sets a goal to capture basic understanding of a device (the composite) by use of a rigorous mathematical model; the second attempts to capture the expertise about the process of fabricating a composite (to date) at a surface level typically expressed in a rule based system. From an AI perspective, these two research lines are attacking distinctly different problems, and both tracks have current limitations. The mathematical modeling approach has yielded a wealth of data but a large number of simplifying assumptions are needed to make numerical simulation tractable. Likewise, although surface level expertise about how to build a particular composite may yield important results, recent trends in the KBS area are towards augmenting surface level problem solving with deeper level knowledge. Many of the relative advantages of composites, e.g., the strength:weight ratio, is most prominent when the entire component is designed as a unitary piece. The bottleneck in undertaking such unitary design lies in the difficulty of the re-design task. Designing the fabrication protocols for a complex-shaped, thick section composite are currently very difficult. It is in fact this difficulty that our research will address.

Punch, W. F., III↗

NASA NTRS1993

New developments in the method of space-time conservation element and solution element: Applications to the Euler and Navier-Stokes equations

A new numerical framework for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods--i.e., finite difference, finite volume, finite element, and spectral methods. It is conceptually simple and designed to avoid several key limitations to the above traditional methods. An explicit model scheme for solving a simple 1-D unsteady convection-diffusion equation is constructed and used to illuminate major differences between the current method and those mentioned above. Unexpectedly, its amplification factors for the pure convection and pure diffusion cases are identical to those of the Leapfrog and the DuFort-Frankel schemes, respectively. Also, this explicit scheme and its Navier-Stokes extension have the unusual property that their stabilities are limited only by the CFL condition. Moreover, despite the fact that it does not use any flux-limiter or slope-limiter, the Navier-Stokes solver is capable of generating highly accurate shock tube solutions with shock discontinuities being resolved within one mesh interval. An accurate Euler solver also is constructed through another extension. It has many unusual properties, e.g., numerical diffusion at all mesh points can be controlled by a set of local parameters.

Chang, Sin-Chung↗