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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: An eigenvalue analysis

A hyperbolic initial-boundary-value problem can be approximated by a system of ordinary differential equations (ODEs) by replacing the spatial derivatives by finite-difference approximations. The resulting system of ODEs is called a semidiscrete approximation. A complication is the fact that more boundary conditions are required for the spatially discrete approximation than are specified for the partial differential equation. Consequently, additional numerical boundary conditions are required and improper treatment of these additional conditions can lead to instability. For a linear initial-boundary-value problem (IBVP) with homogeneous analytical boundary conditions, the semidiscrete approximation results in a system of ODEs of the form du/dt = Au whose solution can be written as u(t) = exp(At)u(O). Lax-Richtmyer stability requires that the matrix norm of exp(At) be uniformly bounded for O less than or = t less than or = T independent of the spatial mesh size. Although the classical Lax-Richtmyer stability definition involves a conventional vector norm, there is no known algebraic test for the uniform boundedness of the matrix norm of exp(At) for hyperbolic IBVPs. An alternative but more complicated stability definition is used in the theory developed by Gustafsson, Kreiss, and Sundstrom (GKS). The two methods are compared.

Warming, Robert F.↗
NASA NTRS1995

Numerical Stability In Hyperbolic Boundary-Value Problems

Technical memorandum discusses stability of numerical solutions involving semidiscrete approximations to hyperbolic partial differential equations in initial-and-boundary-value problems. Topic of practical significance because hyperbolic partial differential equations arise in mathematical modeling of waves and blasts. Solutions often needed over restricted regions of unbounded spaces. Outer boundaries artificial, introduced only to limit domains of numerical solutions. Conditions at such artificial boundaries cause numerical instabilities that degrade computed solutions.

Warming, Robert F.↗
NASA NTRS1983

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems

New convenient stability criteria are provided in this paper for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter plane x or = 0, t or = 0. Using the new criteria, stability is easily established for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in recent literature.

Goldberg, M.↗
NASA NTRS1985

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems

New convenient stability criteria are provided in this paper for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter plane x or = 0, t or = 0. Using the new criteria, stability is easily established for numerous combinations of well known basic schemes and boundary conditins, thus generalizing many special cases studied in recent literature.

Goldberg, M.↗
NASA NTRS1985

New stability criteria for difference approximations of hyperbolic initial-boundary value problems

New convenient stability criteria are provided for a large class of finite difference approximations to initial-boundary value problems associated with the hyperbolic system u sub t = Au sub x + Bu + f in the quarter plane x greater than or equal to 0, t greater than or equal to 0. The criteria are used to easily establish stability for numerous combinations of well known basic schemes and boundary conditions, thus generalizing many special cases studied in the recent literature. A number of examples are examined, including the unitary unconditionally stable Crank-Nicholson scheme and an almost-dissipative unconditionally stable backward Euler scheme.

Goldberg, M.↗
NASA NTRS1986

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems

The purpose of this paper is to achieve more versatile, convenient stability criteria for a wide class of finite-difference approximations to initial boundary value problems associated with the hyperbolic system u sub t = au sub x + Bu + f in the quarter-plane x greater than or equal to 0, t greater than or equal to 0. With these criteria, stability is easily established for a large number of examples, thus incorporating and generalizing many of the cases studied in recent literature.

Goldberg, M.↗
DOE OSTI2018

Bounded Solutions of Hyperbolic Boundary-Value Problems

We investigate linear boundary-value problems for the first-order one-dimensional hyperbolic systems in a strip and establish conditions for the existence and uniqueness of bounded continuous solutions. For this purpose, we suppose that the nondiagonal part of the zero-order coefficients vanish at infinity. Moreover, we establish a dissipativity condition in terms of the boundary data and determine the diagonal part of the zero-order coefficients.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1988

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: Stationary modes

Spatially discrete difference approximations for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline approximation can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.

Warming, Robert F.↗
NASA NTRS1990

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Gottlieb, David↗
NASA NTRS1991

The CFL condition for spectral approximations to hyperbolic initial-boundary value problems

The stability of spectral approximations to scalar hyperbolic initial-boundary value problems with variable coefficients are studied. Time is discretized by explicit multi-level or Runge-Kutta methods of order less than or equal to 3 (forward Euler time differencing is included), and spatial discretizations are studied by spectral and pseudospectral approximations associated with the general family of Jacobi polynomials. It is proved that these fully explicit spectral approximations are stable provided their time-step, delta t, is restricted by the CFL-like condition, delta t less than Const. N(exp-2), where N equals the spatial number of degrees of freedom. We give two independent proofs of this result, depending on two different choices of approximate L(exp 2)-weighted norms. In both approaches, the proofs hinge on a certain inverse inequality interesting for its own sake. The result confirms the commonly held belief that the above CFL stability restriction, which is extensively used in practical implementations, guarantees the stability (and hence the convergence) of fully-explicit spectral approximations in the nonperiodic case.

Gottlieb, David↗
NASA NTRS1987

Convenient stability criteria for difference approximations of hyperbolic initial-boundary value problems. II

The results of Goldberg and Tadmor (1985) are extended to achieve improved stability criteria for a large class of approximations to the initial boundary value problem associated with a particular hyperbolic system in a quarter plane. In a stability analysis, it is shown that the entire approximation is stable if and only if the scalar outflow components of its principal part are stable. Thus, the global stability question is reduced to that of a scalar, homogeneous outflow problem. The stability criteria for the reduced problem, which depend both on the basic scheme and the boundary conditions, but very little on the interaction between the two, are stated and used to establish previous examples and new ones, including a host of dissipative and nondissipative examples. There is no difficulty in extending the stability criteria to two-boundary problems and initial-boundary value problems with variable coefficients.

Goldberg, Moshe↗
NASA NTRS1988

Stability of semi-discrete approximations for hyperbolic initial-boundary-value problems. II - Asymptotic estimates

The stability of semi-discrete approximations for hyperbolic initial-boundary-value problems is considered. Asymptotic estimates of the eigenvalues of the finite-domain problem are given. These estimates are used to examine questions relating the normal mode analysis of the finite-domain problem and the normal mode quarter-plane analysis of the Gustafsson, Kreiss, and Sundstroem theory.

Warming, Robert F.↗
NASA NTRS1994

Fourth order difference methods for hyperbolic IBVP's

Fourth order difference approximations of initial-boundary value problems for hyperbolic partial differential equations are considered. We use the method of lines approach with both explicit and compact implicit difference operators in space. The explicit operator satisfies an energy estimate leading to strict stability. For the implicit operator we develop boundary conditions and give a complete proof of strong stability using the Laplace transform technique. We also present numerical experiments for the linear advection equation and Burgers' equation with discontinuities in the solution or in its derivative. The first equation is used for modeling contact discontinuities in fluid dynamics, the second one for modeling shocks and rarefaction waves. The time discretization is done with a third order Runge-Kutta TVD method. For solutions with discontinuities in the solution itself we add a filter based on second order viscosity. In case of the non-linear Burger's equation we use a flux splitting technique that results in an energy estimate for certain different approximations, in which case also an entropy condition is fulfilled. In particular we shall demonstrate that the unsplit conservative form produces a non-physical shock instead of the physically correct rarefaction wave. In the numerical experiments we compare our fourth order methods with a standard second order one and with a third order TVD-method. The results show that the fourth order methods are the only ones that give good results for all the considered test problems.

Gustafsson, Bertil↗
DOE OSTI2021

Optimizing multigrid reduction-in-time and Parareal coarse-grid operators for linear advection

Parallel-in-time methods, such as multigrid reduction-in-time (MGRIT) and Parareal, provide an attractive option for increasing concurrency when simulating time-dependent partial differential equations (PDEs) in modern high-performance computing environments. While these techniques have been very successful for parabolic equations, it has often been observed that their performance suffers dramatically when applied to advection-dominated problems or purely hyperbolic PDEs using standard rediscretization approaches on coarse grids. In this paper, we apply MGRIT or Parareal to the constant-coefficient linear advection equation, appealing to existing convergence theory to provide insight into the typically nonscalable or even divergent behavior of these solvers for this problem. To overcome these failings, we replace rediscretization on coarse grids with improved coarse-grid operators that are computed by applying optimization techniques to approximately minimize error estimates from the convergence theory. Therefore, one of our main findings is that, in order to obtain fast convergence as for parabolic problems, coarse-grid operators should take into account the behavior of the hyperbolic problem by tracking the characteristic curves. Our approach is tested for schemes of various orders using explicit or implicit Runge–Kutta methods combined with upwind-finite-difference spatial discretizations. In all cases, we obtain scalable convergence in just a handful of iterations, with parallel tests also showing significant speed-ups over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2011

Distributions of Orbital Elements for Meteoroids on Near-Parabolic Orbits According to Radar Observational Data

Some results of the International Heliophysical Year (IHY) Coordinated Investigation Program (CIP) number 65 Meteors in the Earth Atmosphere and Meteoroids in the Solar System are presented. The problem of hyperbolic and near-parabolic orbits is discussed. Some possibilities for the solution of this problem can be obtained from the radar observation of faint meteors. The limiting magnitude of the Kharkov, Ukraine, radar observation program in the 1970 s was +12, resulting in a very large number of meteors being detected. 250,000 orbits down to even fainter limiting magnitude were determined in the 1972-78 period in Kharkov (out of them 7,000 are hyperbolic). The hypothesis of hyperbolic meteors was confirmed. In some radar meteor observations 1 10% of meteors are hyperbolic meteors. Though the Advanced Meteor Orbit Radar (AMOR, New Zealand) and Canadian Meteor Orbit Radar (CMOR, Canada) have accumulated millions of meteor orbits, there are difficulties in comparing the radar observational data obtained from these three sites (New Zealand, Canada, Kharkov). A new global program International Space Weather Initiative (ISWI) has begun in 2010 (http://www.iswi-secretariat.org). Today it is necessary to create the unified radar catalogue of nearparabolic and hyperbolic meteor orbits in the framework of the ISWI, or any other different way, in collaboration of Ukraine, Canada, New Zealand, the USA and, possibly, Japan. Involvement of the Virtual Meteor Observatory (Netherlands) and Meteor Data Centre (Slovakia) is desirable too. International unified radar catalogue of near-parabolic and hyperbolic meteor orbits will aid to a major advance in our understanding of the ecology of meteoroids within the Solar System and beyond.

Kolomiyets, S. V.↗