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NASA NTRS1982

A natural conservative flux difference splitting for the hyperbolic systems of gasdynamics

Gasdynamic hyperbolic systems are treated by a novel conservative flux difference splitting upwind method which is applicable to explicit or implicit and iterative or direct schemes, for marching in time or space with the Euler or compressible Navier-Stokes equations. The method is able to capture sharp shocks correctly by maintaining global conservation. The embedded characteristics formulation is cast in the volumetric internal energy primitive variable, which is effective for real as well as perfect gases. Higher order upwind formulae are constructed from the distribution of pieces of the simple internal flux difference splitting, and the method reverts to first order at the appropriate points in order not to violate the domain of dependence by differencing across discontinuities. At insignificant computational overhead, switching is done by simple algebraic extensions of the truth functions that determine upwind direction for the first order scheme.

Lombard, C. K.↗
NASA NTRS1979

Energy conserving norms for the solution of hyperbolic systems of partial differential equations

The hyperbolic system of partial differential equations with a real constant square coefficient matrix A is considered. The problem of finding an energy conserving norm for the solution of the system is reduced to the problem of characterizing those matrices appearing in the boundary conditions which satisfy two specific matrix equations. Necessary and sufficient conditions on the coefficient matrix A and the matrices appearing in boundary conditions are derived for an energy conserving norm. The conditions serve as criteria on a given system which determine whether or not the solution will have its energy conserved in some norm. Examples of specific systems and boundary conditions are also provided.

Gunzburger, M. D.↗
NASA NTRS1977

On the stability of Galerkin methods for initial-boundary value problems for hyperbolic systems

The stability of approximating the solution of mixed initial-boundary value problems for hyperbolic systems by semidiscrete Galerkin methods is studied. It is shown that a particular straightforward Galerkin method yields an unstable approximation, and that this numerical instability is caused by an improper treatment of the boundary. Stable schemes are then presented, one of which differs from the unstable scheme only insofar as the treatment of the boundary is concerned. These stable schemes make use of a particular matrix which symmetrizes the differential system. It is therefore shown that the use of this matrix is crucial to the stability of the computations as well as for obtaining a priori bounds on the energy of the continuous system. This symmetrizing matrix is also related to the diagonalizing matrix for the system of hyperbolic equations and to the Liapunov matrix for the system of ordinary differential equations resulting from the application of Galerkin's method.

Gunzburger, M. D.↗
NASA NTRS1981

Unconditional instability of inflow dependent boundary conditions in difference approximations to hyperbolic systems

The stability of finite difference approximations to initial boundary hyperbolic systems is studied. As is well known, a proper specification of boundary conditions for such systems is essential for their solutions to be well defined. A discrete analogue of the above is proved - if the numerical boundary conditions are consistent with an inflow part of the problem, they render the overall computation unstable. An example of the inviscid gasdynamics equations is considered.

Tadmor, E.↗
NASA NTRS1981

Determining solutions of hyperbolic systems from incomplete data

An investigation is conducted regarding first-order hyperbolic systems of partial differential equations, taking into account problems for which complete initial data are not available. Problems of the considered kind arise in geophysical applications where satellites are used to collect data. In global weather prediction, it is possible to derive atmospheric temperature and pressure reasonably well over the whole globe from satellite measurements; obtaining the wind field globally is more difficult. It is pointed out that a simple model of atmospheric flow investigated in numerical weather prediction is governed by the shallow water equations. The effect of the Coriolis term on the linearized shallow-water equations is studied.

Bube, K. P.↗
NASA NTRS1980

The Split Coefficient Matrix method for hyperbolic systems of gasdynamic equations

The Split Coefficient Matrix (SCM) finite difference method for solving hyperbolic systems of equations is presented. This new method is based on the mathematical theory of characteristics. The development of the method from characteristic theory is presented. Boundary point calculation procedures consistent with the SCM method used at interior points are explained. The split coefficient matrices that define the method for steady supersonic and unsteady inviscid flows are given for several examples. The SCM method is used to compute several flow fields to demonstrate its accuracy and versatility. The similarities and differences between the SCM method and the lambda-scheme are discussed.

Chakravarthy, S. R.↗
NASA NTRS1982

Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics for real and perfect gases

A conservative flux difference splitting is presented for the hyperbolic systems of gasdynamics. The stable robust method is suitable for wide application in a variety of schemes, explicit or implicit, iterative or direct, for marching in either time or space. The splitting is modeled on the local quasi one dimensional characteristics system for multi-dimensional flow similar to Chakravarthy's nonconservative split coefficient matrix method; but, as the result of maintaining global conservation, the method is able to capture sharp shocks correctly. The embedded characteristics formulation is cast in a primitive variable the volumetric internal energy (rather than the pressure) that is effective for treating real as well as perfect gases. Finally the relationship of the splitting to characteristics boundary conditions is discussed and the associated conservative matrix formulation for a computed blown wall boundary condition is developed as an example. The theoretical development employs and extends the notion of Roe of constructing stable upwind difference formulae by sending split simple one sided flux difference pieces to appropriate mesh sites. The developments are also believed to have the potential for aiding in the analysis of both existing and new conservative difference schemes.

Lombard, C. K.↗
NASA NTRS1976

An implicit finite-difference algorithm for hyperbolic systems in conservation-law form

An implicit finite-difference scheme is developed for the efficient numerical solution of nonlinear hyperbolic systems in conservation-law form. The algorithm is second-order time-accurate, noniterative, and in a spatially factored form. Second- or fourth-order central and second-order one-sided spatial differencing are accommodated within the solution of a block tridiagonal system of equations. Significant conceptual and computational simplifications are made for systems whose flux vectors are homogeneous functions (of degree one), e.g., the Eulerian gasdynamic equations. Conservative hybrid schemes, which switch from central to one-sided spatial differencing whenever the local characteristic speeds are of the same sign, are constructed to improve the resolution of weak solutions. Numerical solutions are presented for a nonlinear scalar model equation and the two-dimensional Eulerian gasdynamic equations.

Beam, R. M.↗
NASA NTRS1976

A rapid solver for hyperbolic systems of equations

A numerical method is proposed for solving the time-dependent compressible Navier-Stokes equations in two dimensions. The equations are time-split into a hyperbolic part and a parabolic part. This paper describes an explicit numerical method for solving the hyperbolic (inviscid) part. The hyperbolic operator is explicit, conservative, uses characteristic relations to predict convection and pressure fields, and is stable under a specified condition.

Maccormack, R. W.↗
NASA NTRS1982

Conservative supra-characteristics method for splitting the hyperbolic systems of gasdynamics with computed boundaries for real and perfect gases

Implicit methods developed by Beam and Warming (1978 and Briley and McDonald (1977) make it possible to overcome the hyperbolic stiffness of the conservative compressible Navier-Stokes equations in the fine wall region computational mesh for high Reynold's number flow. Certain difficulties related to the use of these methods could be overcome by employing an approach reported by Roe (1981). In the present investigation Roe's conceptual framework has been adopted for constructing globally conservative finite difference methods. A globally conservative upwind finite difference method (CSCM) consisting of both implicit interior point and boundary point equations is constructed from a new characteristics based flux difference splitting. It is found that the employed upwind eigenvector split scheme which combines fully coupled implicit interior point and boundary point approximations has the desired properties of robust stability and accuracy.

Lombard, C. K.↗
NASA NTRS1980

On the practical use of high-order methods for hyperbolic systems

The paper tests a number of high order methods on a variety of dynamic problems in one, two, and three space dimensions. The problems covered include wave propagation phenomena as well as an asymptotic approach to a steady state. Consideration is given to both smooth and shocked flows. It is shown that the methods compared require only minor modifications of many existing second-order schemes. Further, the results show that significant gains can be expected from the use of fourth-order methods. Finally, spectral methods are also considered for some of the problems presented.

Turkel, E.↗
NASA NTRS1981

The choice of numerical boundary conditions for hyperbolic systems

Two fundamental problems for mixed initial boundary value problems with applications in fluid mechanics are discussed. First, different stability properties are discussed which are of importance for long time integrations and steady state calculations. Secondly, a new numerical technique for problems with an artificial boundary is introduced.

Gustafson, B.↗
NASA NTRS1978

On the construction and application of implicit factored schemes for conservation laws

Efficient, noniterative, implicit finite difference algorithms are systematically developed for nonlinear conservation laws including purely hyperbolic systems and mixed hyperbolic parabolic systems. Utilization of a rational fraction or Pade time differencing formulas, yields a direct and natural derivation of an implicit scheme in a delta form. Attention is given to advantages of the delta formation and to various properties of one- and two-dimensional algorithms.

Warming, R. F.↗
NASA NTRS1972

Automatic lightning location system

Hyperbolic triangulation method was used for locating lightning storm path and position from VHF lightning charge emissions. Possible applications in electric power companies, forest fire lookout centers, airports, and pipeline companies are indicated.

Holdsworth, T.↗
NASA NTRS1976

The linear quadratic optimal control problem for infinite dimensional systems over an infinite horizon - Survey and examples

Available published results are surveyed for a special class of infinite-dimensional control systems whose evolution is characterized by a semigroup of operators of class C subscript zero. Emphasis is placed on an approach that clarifies the system-theoretic relationship among controllability, stabilizability, stability, and the existence of a solution to an associated operator equation of the Riccati type. Formulation of the optimal control problem is reviewed along with the asymptotic behavior of solutions to a general system of equations and several theorems concerning L2 stability. Examples are briefly discussed which involve second-order parabolic systems, first-order hyperbolic systems, and distributed boundary control.

Bensoussan, A.↗
NASA NTRS1977

An accuracy analysis of the LDAR system

An accuracy report of the Lightning Detection and Ranging System is presented. The effect of quantizing errors are modelled by use of a computer. The errors in the four configurations of the LDAR System are compared to the limiting errors in an ideal hyperbolic system. Performance data from the track of a jet plane and for the indicated position of a fixed lightning simulator are analyzed for dispersions in the data. Error models show the quality and the areas of highly accurate data, and show how the data deteriorates outside the primary measuring range.

Poehler, H. A.↗