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NASA NTRS1986

The discrete one-sided Lipschitz condition for convex scalar conservation laws

Physical solutions to convex scalar conservation laws satisfy a one-sided Lipschitz condition (OSLC) that enforces both the entropy condition and their variation boundedness. Consistency with this condition is therefore desirable for a numerical scheme and was proved for both the Godunov and the Lax-Friedrichs scheme--also, in a weakened version, for the Roe scheme, all of them being only first order accurate. A new, fully second order scheme is introduced here, which is consistent with the OSLC. The modified equation is considered and shows interesting features. Another second order scheme is then considered and numerical results are discussed.

Brenier, Yann↗

NASA NTRS1987

Conservative zonal schemes for patched grids in 2 and 3 dimensions

The computation of flow over complex geometries, such as realistic aircraft configurations, poses difficult grid generation problems for computational aerodynamicists. The creation of a traditional, single-module grid of acceptable quality about an entire configuration may be impossible even with the most sophisticated of grid generation techniques. A zonal approach, wherein the flow field is partitioned into several regions within which grids are independently generated, is a practical alternative for treating complicated geometries. This technique not only alleviates the problems of discretizing a complex region, but also facilitates a block processing approach to computation thereby circumventing computer memory limitations. The use of such a zonal scheme, however, requires the development of an interfacing procedure that ensures a stable, accurate, and conservative calculation for the transfer of information across the zonal borders.

Hessenius, Kristin A.↗

NASA NTRS1987

A coupled-mode theory for multiwaveguide systems satisfying the reciprocity theorem and power conservation

Two sets of coupled-mode equations for multiwaveguide systems are derived using a generalized reciprocity relation; one set for a lossless system, and the other for a general lossy or lossless system. The second set of equations also reduces to those of the first set in the lossless case under the condition that the transverse field components are chosen to be real. Analytical relations between the coupling coefficients are shown and applied to the coupling of mode equations. It is shown analytically that these results satisfy exactly both the reciprocity theorem and power conservation. New orthogonal relations between the supermodes are derived in matrix form, with the overlap integrals taken into account.

Chuang, Shun-Lien↗

NASA NTRS1987

Conserved variable analysis of the convective boundary layer thermodynamic structure over the tropical oceans

An analysis of FGGE dropwindsonde data using conserved thermodynamic variables shows mixing line structures for the convective boundary layer over the equatorial Pacific. Deeper boundary layers show a double structure. Reversals of the gradients of mixing ratio and equivalent potential temperature above the boundary-layer top are present in all the averages and suggest that the origin of the air sinking into the boundary layer needs further study.

Betts, Alan K.↗

NASA NTRS1987

Semi-implicit and fully implicit shock-capturing methods for hyperbolic conservation laws with stiff source terms

Some numerical aspects of finite-difference algorithms for nonlinear multidimensional hyperbolic conservation laws with stiff nonhomogeneous (source) terms are discussed. If the stiffness is entirely dominated by the source term, a semi-implicit shock-capturing method is proposed provided that the Jacobian of the source terms possesses certain properties. The proposed semi-implicit method can be viewed as a variant of the Bussing and Murman point-implicit scheme with a more appropriate numerical dissipation for the computation of strong shock waves. However, if the stiffness is not solely dominated by the source terms, a fully implicit method would be a better choice. The situation is complicated by problems that are higher than one dimension, and the presence of stiff source terms further complicates the solution procedures for alternating direction implicit (ADI) methods. Several alternatives are discussed. The primary motivation for constructing these schemes was to address thermally and chemically nonequilibrium flows in the hypersonic regime. Due to the unique structure of the eigenvalues and eigenvectors for fluid flows of this type, the computation can be simplified, thus providing a more efficient solution procedure than one might have anticipated.

Yee, H. C.↗

NASA NTRS1987

A Taylor weak-statement algorithm for hyperbolic conservation laws

Finite element analysis, applied to computational fluid dynamics (CFD) problem classes, presents a formal procedure for establishing the ingredients of a discrete approximation numerical solution algorithm. A classical Galerkin weak-statement formulation, formed on a Taylor series extension of the conservation law system, is developed herein that embeds a set of parameters eligible for constraint according to specification of suitable norms. The derived family of Taylor weak statements is shown to contain, as special cases, over one dozen independently derived CFD algorithms published over the past several decades for the high speed flow problem class. A theoretical analysis is completed that facilitates direct qualitative comparisons. Numerical results for definitive linear and nonlinear test problems permit direct quantitative performance comparisons.

Baker, A. J.↗

NASA NTRS1986

Breakdown of the conservative potential equation

The breakdown of the conservative potential approximation occurs for all the airfoils tested. It develops as soon as the shock waves appear in the flow field. Since shock waves are not properly represented by the potential approximation, it is conjectured that the breakdown is due to the isentropic shock jump condition of the potential approximation.

Salas, M. D.↗

NASA NTRS1988

A study of numerical methods for hyperbolic conservation laws with stiff source terms

The proper modeling of nonequilibrium gas dynamics is required in certain regimes of hypersonic flow. For inviscid flow this gives a system of conservation laws coupled with source terms representing the chemistry. Often a wide range of time scales is present in the problem, leading to numerical difficulties as in stiff systems of ordinary differential equations. Stability can be achieved by using implicit methods, but other numerical difficulties are observed. The behavior of typical numerical methods on a simple advection equation with a parameter-dependent source term was studied. Two approaches to incorporate the source term were utilized: MacCormack type predictor-corrector methods with flux limiters, and splitting methods in which the fluid dynamics and chemistry are handled in separate steps. Various comparisons over a wide range of parameter values were made. In the stiff case where the solution contains discontinuities, incorrect numerical propagation speeds are observed with all of the methods considered. This phenomenon is studied and explained.

Leveque, R. J.↗

NASA NTRS1988

Generalized energy and potential enstrophy conserving finite difference schemes for the shallow water equations

The conditions under which finite difference schemes for the shallow water equations can conserve both total energy and potential enstrophy are considered. A method of deriving such schemes using operator formalism is developed. Several such schemes are derived for the A-, B- and C-grids. The derived schemes include second-order schemes and pseudo-fourth-order schemes. The simplest B-grid pseudo-fourth-order schemes are presented.

Abramopoulos, Frank↗

NASA NTRS1988

'Flux conservation' by a Schwarzschild gravitational lens

It is demonstrated, by an explicit calculation, that a single, isolated Schwarzschild gravitational lens leads to a vanishing net total amplification on its own, without any 'help' from other lenses. It is shown that, in the limit of small M/R (where M is the mass of the lens in geometrical units and R is the distance from the source to the observers), the positive net total amplification, obtained from the classical approximation of the amplification for small impact angle lensing, is balanced by a negative net total amplification from the first-order term in M/R in an approximation of the amplification for large impact angle lensing derived here. The relation of these results to the common interpretation of average flux conservation by gravitational lenses is discussed as due to the collective action of many lenses in a cosmological framework.

Shulami, I.↗

NASA NTRS1988

Non-oscillatory central differencing for hyperbolic conservation laws

Many of the recently developed high resolution schemes for hyperbolic conservation laws are based on upwind differencing. The building block for these schemes is the averaging of an appropriate Godunov solver; its time consuming part involves the field-by-field decomposition which is required in order to identify the direction of the wind. Instead, the use of the more robust Lax-Friedrichs (LxF) solver is proposed. The main advantage is simplicity: no Riemann problems are solved and hence field-by-field decompositions are avoided. The main disadvantage is the excessive numerical viscosity typical to the LxF solver. This is compensated for by using high-resolution MUSCL-type interpolants. Numerical experiments show that the quality of results obtained by such convenient central differencing is comparable with those of the upwind schemes.

Nessyahu, Haim↗

NASA NTRS1988

Some observations on boundary conditions for numerical conservation laws

Four choices of outflow boundary conditions are considered for numerical conservation laws. All four methods are stable for linear problems, for which examples are presented where either a boundary layer forms or the numerical scheme, together with the boundary condition, is unstable due to the formation of a reflected shock. A simple heuristic argument is presented for determining the suitability of the boundary condition.

Kamowitz, David↗

NASA NTRS1989

Conservation equations and physical models for hypersonic air flows in thermal and chemical nonequilibrium

The conservation equations for simulating hypersonic flows in thermal and chemical nonequilibrium and details of the associated physical models are presented. These details include the curve fits used for defining thermodynamic properties of the 11 species air model, curve fits for collision cross sections, expressions for transport properties, the chemical kinetics models, and the vibrational and electronic energy relaxation models. The expressions are formulated in the context of either a two or three temperature model. Greater emphasis is placed on the two temperature model in which it is assumed that the translational and rotational energy models are in equilibrium at the translational temperature, T, and the vibrational, electronic, and electron translational energy modes are in equilibrium at the vibrational temperature, T sub v. The eigenvalues and eigenvectors associated with the Jacobian of the flux vector are also presented in order to accommodate the upwind based numerical solutions of the complete equation set.

Peter A. Gnoffo↗

NASA NTRS1989

Second-order accurate nonoscillatory schemes for scalar conservation laws

Explicit finite difference schemes for the computation of weak solutions of nonlinear scalar conservation laws is presented and analyzed. These schemes are uniformly second-order accurate and nonoscillatory in the sense that the number of extrema of the discrete solution is not increasing in time.

Huynh, Hung T.↗

NASA NTRS1988

A sufficient condition for the stability of conservative gyroscopic systems

A sufficient condition for the stability of conservative gyroscopic systems with negative definite stiffness is presented. The conditions for stability are stated in terms of the definiteness of certain combinations of the coefficient matrices of the equations of motion. These conditions yield design constraints in terms of the physical parameters of the system. An example is given to illustrate the correctness of the results, as well as to provide a comparison with the results of other researchers.

Inman, D. J.↗

NASA NTRS1989

Universal limiter for high order explicit conservative advection schemes

A simple method for construction of nonoscillatory, explicit, conservative advection schemes of arbitrarily high accuracy is described. Four test profiles are considered: an isolated sine-squared wave, a unit step function, a semiellipse, and a narrow Gaussian. A universal limiter guarantees monotonic resolution of the step. Sharpness increases uniformly with the order of the base scheme.

Leonard, B. P.↗

NASA NTRS1989

Ozone conservation and entrainment in cumulus congestus

This study demonstrates that ozone mixing ratio is conserved during moist convection and can be used as a tracer for cloud entrainment studies. The approach is to apply mixing line analysis to pairs of liquid water potential temperature, total water mixing ratio, O3 and pseudo-equivalent potential temperature derived from aircraft penetrations of growing cumulus congestus. Conclusions about entrainment from the mixing diagrams employing O3 agree with those using thermodynamic quantities. Any disagreement uncovered deficiencies in the water substance measurement technique.

Pearson, R., Jr.↗

NASA NTRS1990

Triangle based TVD schemes for hyperbolic conservation laws

A triangle based total variation diminishing (TVD) scheme for the numerical approximation of hyperbolic conservation laws in two space dimensions is constructed. The novelty of the scheme lies in the nature of the preprocessing of the cell averaged data, which is accomplished via a nearest neighbor linear interpolation followed by a slope limiting procedures. Two such limiting procedures are suggested. The resulting method is considerably more simple than other triangle based non-oscillatory approximations which, like this scheme, approximate the flux up to second order accuracy. Numerical results for linear advection and Burgers' equation are presented.

Durlofsky, Louis J.↗