%A T. A. Zang %A M. Y. Hussaini %T A three-dimensional spectral algorithm for simulations of transition and turbulence %R ICASE Report No. 85-19 %I NASA Langley Research Centre %D March 1985 %P 39 %O AIAA Paper No. 85-0296 presented at the AIAA 23rd Aerospace Sciences Meeting, January 14-17, 1985, Reno, Nevada. %X A spectral algorithm for simulating three-dimensional, incompressible, parallel shear flows is described. It applies to the channel, to the parallel boundary layer, and to other shear flows with one wall-bounded and two periodic directions. Representative applications to the channel and to the heated boundary layer are presented. %A J. P. Drummond %A M. Y. Hussaini %A T. A. Zang %T Spectral methods for modeling supersonic chemically reacting flow fields %R ICASE Report No. 85-20 %I NASA Langley Research Centre %D March 1985 %P 37 %X A numerical algorithm has been developed for solving the equations describing chemically reacting supersonic flows. The algorithm employs a two- stage Runge-Kutta method for integrating the equations in time and a Chebyshev spectral method for integrating the equations in space. The accuracy and efficiency of the technique have been assessed by comparison with an existing implicit finite-difference procedure for modeling chemically reacting flows. The comparison showed that the new procedure yielded equivalent accuracy on much coarser grids as compared to the finite-difference procedure with resultant significant gains in computational efficiency. %A R. J. LeVeque %T Intermediate boundary conditions for LOD, ADI, and approximate factorization methods %R ICASE Report No. 85-21 %I NASA Langley Research Centre %D April 1985 %P 24 %O Submitted to J. Comput. Phys. %X A general approach to determining the correct intermediate boundary conditions for dimensional splitting methods is presented and illustrated. The intermediate solution U* is viewed as a second-order accurate approximation to a modified equation. Deriving the modified equation and using the relationship between this equation and the original equation allows us to determine the correct boundary conditions for U*. To illustrate this technique, we apply it to LOD and ADI methods for the heat equation in two and three space dimensions. The approximate factorization method is considered in slightly more generality. %A I. G. Rosen %T Spline-based Rayleigh-Ritz methods for the approximation of the natural modes of vibration for flexible beams with tip bodies %R ICASE Report No. 85-22 %I NASA Langley Research Centre %D March 1985 %P 31 %O Submitted to Quarterly of Applied Mathematics. %X Rayleigh-Ritz methods for the approximation of the natural modes for a class of vibration problems involving flexible beams with tip bodies using subspaces of piecewise polynomial spline functions are developed. An abstract operator theoretic formulation of the eigenvalue problem is derived and spectral properties investigated. The existing theory for spline-based Rayleigh-Ritz methods applied to elliptic differential operators and the approximation properties of interpolatory splines are used to argue convergence and establish rates of convergence. An example and numerical results are discussed. %A A. Bayliss %A L. Maestrello %A P. Parikh %A and E. Turkel %T Numerical simulation of boundary layer excitation by surface heating/cooling %R ICASE Report No. 85-23 %I NASA Langley Research Centre %D March 1985 %P 22 %O also as AIAA Paper No. 85-0565, AIAA Shear Flow Control Conference, March 12-14, 1985, Boulder, CO. %X This paper is a numerical study of the concept of active control of growing disturbances in an unstable compressible flow by using time periodic, localized surface heating. The simulations are calculated by a fourth-order accurate solution of the compressible, laminar Navier-Stokes equations. Fourth-order accuracy is particularly important for this problem because the solution must be computed over many wavelengths. The numerical results demonstrate the growth of an initially small fluctuation into the nonlinear regime where a local breakdown into smaller scale disturbances can be observed. It is shown that periodic surface heating over a small strip can reduce the level of the fluctuation provided that the phase of the heating current is properly chosen. %A M. Hossain %A G. Vahala %A D. Montgomery %T Forced MHD turbulence in a uniform external magnetic field %R ICASE Report No. 85-24 %I NASA Langley Research Centre %D March 1985 %D 39 %O Submitted to Phys. Fluid. %X Two-dimensional dissipative MHD turbulence is randomly driven at small spatial scales and is studied by numerical simulation in the presence of a strong uniform external magnetic field. A novel behavior is observed which is apparently distinct from the inverse cascade which prevails in the absence of an external magnetic field. The magnetic spectrum becomes dominated by the three longest-wavelength Alfv'n waves in the system allowed by the boundary conditions: those which, in a box size of edge 2 pi, have wave numbers (kx, ky) = (1, 0), (1, 1), and (1, -1), where the external magnetic field is in the x direction. At any given instant, one of these three modes dominates the vector potential spectrum, but they do not constitute a resonantly coupled triad. Rather, they are apparently coupled by the smaller- scale turbulence. %A S. F. Davis %T Shock capturing %R ICASE Report No. 85-25 %I NASA Langley Research Centre %D April 1985 %P 23 %O To appear in Numerical Methods for Partial Differential Equations, (S. I. Hariharan and T. H. Moulder, eds.), Pitman Press, 1986. %X This chapter describes recent developments which have improved our understanding of how finite difference methods resolve discontinuous solutions to hyperbolic partial differential equations. As a result of this understanding improved shock capturing methods are currently being developed and tested. Some of these methods are described and numerical results are presented showing their performance on problems containing shocks in one and two dimensions. We begin this discussion by defining what is meant by a conservative difference scheme and showing that conservation implies that, except in very special circumstances, shocks must be spread over at least two grid intervals. These two interval shocks are actually attained in one dimension if the shock is steady and an upwind scheme is used. By analyzing this case, we determine the reason for this excellent shock resolution and use this result to provide a mechanism for improving the resolution of two-dimensional steady shocks. Unfortunately, this same analysis shows that these results cannot be extended to shocks which move relative to the computing grid. To deal with moving shocks and contact discontinuities we introduce total variation diminishing (TVD) finite difference schemes and flux limiters. We show that TVD schemes are not necessarily upwind, but that upwind TVD schemes perform better because they permit a wider choice of flux limiters. The advantage of non-upwind TVD schemes is that they are easy to implement. Indeed, it is possible to add an appropriately chosen artificial viscosity to a conventional scheme such as MacCormack's method and make it TVD. We conclude by presenting some theoretical results on flux limiters and some numerical computations to illustrate the theory. %A A. Brandt %A S. Ta'asan %T Multigrid solutions to quasi-elliptic schemes. %R ICASE Report No. 85-26 %I NASA Langley Research Centre %D May 1985 %P 21 %O also in Progress and Supercomputing in Computational Fluid Dynamics, (Earl. S. Murman and Saul Abarbanel, eds.), Birkhauser Boston, Inc., (tentative publication date: August 1985). %X Quasi-elliptic schemes arise from central differencing or finite element discretization of elliptic systems with odd order derivatives on non-staggered grids. They are somewhat unstable and less accurate then corresponding staggered-grid schemes. When usual multigrid solvers are applied to them, the asymptotic algebraic convergence is necessarily slow. Nevertheless, it is shown by mode analyses and numerical experiments that the usual FMG algorithm is very efficient in solving quasi-elliptic equations to the level of truncation errors. Also, a new type of multigrid algorithm is presented, mode analyzed and tested, for which even the asymptotic algebraic convergence is fast. The essence of that algorithm is applicable to other kinds of problems, including highly indefinite ones. %A T. A. Zang %A M. Y. Hussaini %T On spectral multigrid methods for the time-dependent Navier-Stokes equations %R ICASE Report No. 85-27 %I NASA Langley Research Centre %D May 1985 %P 24 %O Presented at the 2nd Copper Mountain Conference on Multigrid Methods, April 1-3, 1985, Copper Mountain, CO. %X A new splitting scheme is proposed for the numerical solution of the time-dependent, incompressible Navier-Stokes equations by spectral methods. A staggered grid is used for the pressure, improved intermediate boundary conditions are employed in the split step for the velocity, and spectral multigrid techniques are used for the solution of the implicit equations. %A S. Osher %A E. Tadmor %T On the convergence of difference approximations to scalar conservation laws %R ICASE Report No. 85-28 %I NASA Langley Research Centre %D May 1985 %P 70 %O Submitted to Math. Comp. %X We present a unified treatment of explicit in time, two level, second order resolution, total variation diminishing, approximations to scalar conservation laws. The schemes are assumed only to have conservation form and incremental form. We introduce a modified flux and a viscosity coefficient and obtain results in terms of the latter. The existence of a cell entropy inequality is discussed and such an equality for all entropies is shown to imply that the scheme is an E scheme on monotone (actually more general) data, hence at most only first order accurate in general. Convergence for TVD-SOR schemes approximating convex or concave conservation laws is shown by enforcing a single discrete entropy inequality. %A P. Mehrotra %A J. Van\ Rosendale %T The BLAZE language: A parallel language for scientific programming %R ICASE Report No. 85-29 %I NASA Langley Research Centre %D May 1985 %P 57 %O Submitted to Parallel Computing. %X Programming multiprocessor parallel architectures is a complex task. This paper describes a Pascal-like scientific programming language, Blaze, designed to simplify this task. Blaze contains array arithmetic, "forall" loops, and APL-style accumulation operators, which allow natural expression of fine grained parallelism. It also employs an applicative or functional procedure invocation mechanism, which makes it easy for compilers to extract coarse grained parallelism using machine specific program restructuring. Thus Blaze should allow one to achieve highly parallel execution on multiprocessor architectures, while still providing the user with conceptually sequential control flow. A central goal in the design of Blaze is portability across a broad range of parallel architectures. The multiple levels of parallelism present in Blaze code, in principle, allows a compiler to extract the types of parallelism appropriate for the given architecture, while neglecting the remainder. This paper describes the features of Blaze, and shows how this language would be used in typical scientific programming. %A L. N. Trefethen %A L. Halpern %T Well-Posedness of one-way wave equations and absorbing boundary conditions %R ICASE Report No. 85-30 %I NASA Langley Research Centre %D June 1985 %P 23 %O Submitted to Math. Comp. %X A one-way wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. The construction of such equations can be reduced to the approximation of the square root of 1 - s2 on [-1,1] by a rational function r(s) = Pm(s)/qn(s). This paper characterizes those rational functions r for which the corresponding one-way wave equation is well-posed, both as a partial differential equation and as an absorbing boundary condition for the wave equation. We find that if r(s) interpolates the square root of 1 - s2 at sufficiently many points in (-1,1), then well- posedness is assured. It follows that absorbing boundary conditions based on Pade approximation are well-posed if and only if (m,n) lies in one of two distinct diagonals in the Pade table, the two proposed by Engquist and Majda. Analogous results also hold for one-way wave equations derived from Chebyshev or least-squares approximation. %A G. Majda %T A new theory for multistep discretizations of stiff ordinary differential equations: Stability with large step sizes %R ICASE Report No. 85-31 %I NASA Langley Research Centre %P 70 %O Submitted to Math. Comp. %X In this paper we consider a large set of variable coefficient linear systems of ordinary differential equations which possess two different time scales, a slow one and a fast one. A small parameter epsilon characterizes the stiffness of these systems. We approximate a system of o.d.e.s in this set by a general class of multistep discretizations which includes both one- leg and linear multistep methods. We determine sufficient conditions under which each solution of a multistep method is uniformly bounded, with a bound which is independent of the stiffness of the system of o.d.e.s, when the step size resolves the slow time scale but not the fast one. We call this property stability with large step sizes. The theory presented in this paper lets us compare properties of one-leg methods and linear multistep methods when they approximate variable coefficient systems of stiff o.d.e.s. In particular, we show that one-leg methods have better stability properties with large step sizes than their linear multistep counterparts. This observation is consistent with results obtained by Dahlquist and Lindberg {11}, Nevanlinna and Liniger {32} and van Veldhuizen {41}. Our theory also allows us to relate the concept of D- stability (van Veldhuizen {41}) to the usual notions of stability and stability domains and to the propagation of errors for multistep methods which use large step sizes. %A H. T. Banks %T On a variational approach to some parameter estimation problems %R ICASE Report No. 85-32 %I NASA Langley Research Centre %P 38 %O Invited lecture, Internationl Conference on Control Theory for Distributed Parameter Systems and Applications, July 9 - 14, 1984, Voran, Austria. %X We consider examples (1-D seismic, large flexible structures, bioturbation, nonlinear population dispersal) in which a variational setting can provide a convenient framework for convergence and stability arguments in parameter estimation problems. %A S. I. Hariharan %T Absorbing boundary conditions for exterior problems %R ICASE Report No. 85-33 %I NASA Langley Research Centre %D July 1985 %P 33 %O To appear in Numerical Methods for Partial Differential Equations, (S. I Hariharan and T. H. Moulden, eds.), Pitman Press, 1986. %X In this paper we consider elliptic and hyperbolic problems in unbounded regions. These problems, when one wants to solve them numerically, have the difficulty of prescribing boundary conditions at infinity. Computationally, one needs a finite region in which to solve these problems. The corresponding conditions at infinity imposed on the finite distance boundaries should dictate the boundary condition at infinity and be accurate with respect to the interior numerical scheme. Such boundary conditions are commonly referred to as absorbing boundary conditions. This paper presents a survey and covers our own treatment on these boundary conditions for wave-like equations.