Solution of one-dimensional elastic wave problems by the method of characteristics.
Elastic wave problems involving one space variable solved by hyperbolic partial differential equations
SHOCK DISCONTINUITY↗
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Stability analysis of spectral methods for hyperbolic initial-boundary value systems
A constant coefficient hyperbolic system in one space variable, with zero initial data is discussed. Dissipative boundary conditions are imposed at the two points x = + or - 1. This problem is discretized by a spectral approximation in space. Sufficient conditions under which the spectral numerical solution is stable are demonstrated - moreover, these conditions have to be checked only for scalar equations. The stability theorems take the form of explicit bounds for the norm of the solution in terms of the boundary data. The dependence of these bounds on N, the number of points in the domain (or equivalently the degree of the polynomials involved), is investigated for a class of standard spectral methods, including Chebyshev and Legendre collocations.
Gottlieb, D.↗
Fast wavelet based algorithms for linear evolution equations
A class was devised of fast wavelet based algorithms for linear evolution equations whose coefficients are time independent. The method draws on the work of Beylkin, Coifman, and Rokhlin which they applied to general Calderon-Zygmund type integral operators. A modification of their idea is applied to linear hyperbolic and parabolic equations, with spatially varying coefficients. A significant speedup over standard methods is obtained when applied to hyperbolic equations in one space dimension and parabolic equations in multidimensions.
Engquist, Bjorn↗
Resin-Impregnated Carbon Ablator: A New Ablative Material for Hyperbolic Entry Speeds
Ablative materials are required to protect a space vehicle from the extreme temperatures encountered during the most demanding (hyperbolic) atmospheric entry velocities, either for probes launched toward other celestial bodies, or coming back to Earth from deep space missions. To that effect, the resin-impregnated carbon ablator (RICA) is a high-temperature carbon/phenolic ablative thermal protection system (TPS) material designed to use modern and commercially viable components in its manufacture. Heritage carbon/phenolic ablators intended for this use rely on materials that are no longer in production (i.e., Galileo, Pioneer Venus); hence the development of alternatives such as RICA is necessary for future NASA planetary entry and Earth re-entry missions. RICA s capabilities were initially measured in air for Earth re-entry applications, where it was exposed to a heat flux of 14 MW/sq m for 22 seconds. Methane tests were also carried out for potential application in Saturn s moon Titan, with a nominal heat flux of 1.4 MW/sq m for up to 478 seconds. Three slightly different material formulations were manufactured and subsequently tested at the Plasma Wind Tunnel of the University of Stuttgart in Germany (PWK1) in the summer and fall of 2010. The TPS integrity was well preserved in most cases, and results show great promise.
Esper, Jaime↗
A practical assessment of spectral accuracy for hyperbolic problems with discontinuities
Physical and transform space filtering has been applied to the Fourier spectral collocation solution of the constant coefficient scalar wave equation with a discontinuous initial condition. High order accuracy can be extracted from the unfiltered solution. Smooth, high order Fourier space filtering gives expected polynomial order solutions away from the discontinuity. Spectral accuracy is observed with the physical space filter of Gottlieb and Tadmor.
Kopriva, David A.↗
A practical assessment of spectral accuracy for hyperbolic problems with discontinuities
Physical and transform space filtering has been applied to the Fourier spectral collocation solution of the constant coefficient scalar wave equation with a discontinuous initial condition. High order accuracy can be extracted from the unfiltered solution. Smooth, high order Fourier space filtering gives expected polynomial order solutions away from the discontinuity. Spectral accuracy is observed with the physical space filter of Gottlieb and Tadmor.
Kopriva, David A.↗
Hyperbolic Orbits and the Planetary Flylby Anomaly
Space probes in the Solar System have experienced unexpected changes in velocity known as the flyby anomaly [1], as well as shifts in acceleration referred to as the Pioneer anomaly [2-4]. In the case of Earth flybys, ESA s Rosetta spacecraft experienced the flyby effect and NASA s Galileo and NEAR satellites did the same, although MESSENGER did not possibly due to a latitudinal property of gravity assists. Measurements indicate that both anomalies exist, and explanations have varied from the unconventional to suggestions that new physics in the form of dark matter might be the cause of both [5]. Although dark matter has been studied for over 30 years, there is as yet no strong experimental evidence supporting it [6]. The existence of dark matter will certainly have a significant impact upon ideas regarding the origin of the Solar System. Hence, the subject is very relevant to planetary science. We will point out here that one of the fundamental problems in science, including planetary physics, is consistency. Using the well-known virial theorem in astrophysics, it will be shown that present-day concepts of orbital mechanics and cosmology are not consistent for reasons having to do with the flyby anomaly. Therefore, the basic solution regarding the anomalies should begin with addressing the inconsistencies first before introducing new physics.
Wilson, T.L.↗
Some problems of planetary atmosphere entry.
Space vehicles aerodynamic heating at hyperbolic entry speed, particulary ablation materials
Allen, H. J.↗
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.↗
Triangle based adaptive stencils for the solution of 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.↗
On the global dynamics of adaptive systems - A study of an elementary example
The inherent nonlinear character of adaptive systems poses serious theoretical problems for the analysis of their dynamics. On the other hand, the importance of their dynamic behavior is directly related to the practical interest in predicting such undesirable phenomena as nonlinear oscillations, abrupt transients, intermittence or a high sensitivity with respect to initial conditions. A geometrical/qualitative description of the phase portrait of a discrete-time adaptive system with unmodeled disturbances is given. For this, the motions in the phase space are referred to normally hyperbolic (structurally stable) locally invariant sets. The study is complemented with a local stability analysis of the equilibrium point and periodic solutions. The critical character of adaptive systems under rather usual working conditions is discussed. Special emphasis is put on the causes leading to intermittence. A geometric interpretation of the effects of some commonly used palliatives to this problem is given. The 'dead-zone' approach is studied in more detail. The predicted dynamics are compared with simulation results.
Espana, Martin D.↗
Multigrid Reduction in Time for Chaotic and Hyperbolic Problems (Final Report)
The coming massive parallelism of exascale computing presents a pressing challenge for the many DOE simulations of time-dependent partial differential equations (PDEs), which typically use traditional sequential time stepping methods. Since this traditional approach is inherently serial, it presents a sequential bottleneck when moving to exascale computing, because future performance gains will come through greater concurrency, not faster clock speeds. Thus, the goal of this work is to research parallelism in time, i.e., methods that compute multiple time values simultaneously, not sequentially. The focus will be on hyperbolic and chaotic problems of interest to DOE, with the goal of enabling scalable simulations of time-dependent hyperbolic and chaotic problems on future architectures. The chosen methodology for solving these problems parallel-in-time is multigrid, because multigrid (when it works) is a powerful, optimal, and scalable solver for discretized PDEs. Multigrid is already commonly used in many DOE simulations for scalably and optimally solving space-only PDE problems. The areas of hyperbolic and chaotic problems are chosen because of their relevance to problems of programmatic interest to DOE. However, these problems are also well-known to be difficult for parallel-in-time methods, with the most common method, parareal, diverging in many cases. The current state of-the-art for parallel-in-time at LLNL is the multigrid reduction in time (MGRIT) XBraid package, which also struggles for such problems, while still showing some improvement over parareal. In summary, new methods are needed for an efficient parallel-in-time scheme for hyperbolic and chaotic problems, and this work shall research promising new multigrid methods in this area. In particular, this work shall continue researching the directions from the current collaboration with Dr. Falgout, which are laid out in the work Toward Parallel in Time for Chaotic Dynamical Systems and showed the first known results of a parallel-in-time speedup for a chaotic problem. This work outlines two key improvements to XBraid for chaotic problems, the so-called “theta” and “delta-correction” methods. Here, these two improvements will be further researched and improved (including with a new relaxation method inspired by on Least Squares Shadowing (LSS)) and explored for more complicated problems.
97 MATHEMATICS AND COMPUTING↗
Space vehicle deployment from Space Station
When launching a spacecraft from Earth parking orbit to deep space, it is highly desirable to have the hyperbolic excess velocity vector (v-infinity) contained in the parking orbit plane. Ground launches can force the parking orbit plane to contain the v-infinity vector by using launch azimuth and lift-off time as independent variables. When launching from the Space Station, a new set of variables comes into play. The Space Station orbit is of fixed inclination but precessing due to the Earth's oblateness. Its plane will seldom (and may never) contain the desired v-infinity vector. Consequently, the departure strategy will usually require multiple burns and include a plane change. Also, the concept of 'launch window' will be somewhat different from Earth surface launches. An analysis of the deployment of interplanetary spacecraft from the Space Station is described, with emphasis on the effect of the trajectory characteristics on station operations. Several planetary mission types are analyzed, including manned Mars missions and unmanned high declination departures. The constraint of Space Station orbit nodal position is quantified and the operational implications for Space Station reboost strategy are examined.
Henry, Paul K.↗
Space Vehicle Deployment from Space Station Orbit
When launching a spacecraft from Earth parking orbit to deep space, it is highly desirable to have the hyperbolic excess velocity vector (v-infinity) contained in the parking orbit plane. Ground launches can force the parking orbit plane to contain the v-infinity vector by using launch azimuth and lift-off time as independent variables. When launching from the Space Station, a new set of variables comes into play. The Station orbit is of fixed inclination but precessing due to the Earth's oblateness. Its plane will seldom (and may never) contain the desired v-infinity vector. Consequently, the departure strategy will usually require multiple burns and include a plane change. Also, the concept of "launch period" will be somewhat different from Earth surface launches. An analysis of the deployment of interplanetary spacecraft from Space Station is described, with emphasis on the effect of the trajectory characteristics on station operations. Several planetary mission types are analyzed for manned Mars missions. In addition, high declination departures of spacecraft on unmanned missions to an asteroid are examined. The constraint of Station orbit nodal position is quantified and the operational implications for station reboost strategy are examined.
Henry, Paul K.↗
On a fourth order accurate implicit finite difference scheme for hyperbolic conservation laws. I - Nonstiff strongly dynamic problems
An implicit finite difference method of fourth order accuracy in space and time is introduced for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The basic form of the method is a two-level scheme which is unconditionally stable and nondissipative. The scheme uses only three mesh points at level t and three mesh points at level t + delta t. The dissipative version of the basic method given is conditionally stable under the CFL (Courant-Friedrichs-Lewy) condition. This version is particularly useful for the numerical solution of problems with strong but nonstiff dynamic features, where the CFL restriction is reasonable on accuracy grounds. Numerical results are provided to illustrate properties of the proposed method.
Harten, A.↗
Encke-Beta Predictor for Orion Burn Targeting and Guidance
The state vector prediction algorithm selected for Orion on-board targeting and guidance is known as the Encke-Beta method. Encke-Beta uses a universal anomaly (beta) as the independent variable, valid for circular, elliptical, parabolic, and hyperbolic orbits. The variable, related to the change in eccentric anomaly, results in integration steps that cover smaller arcs of the trajectory at or near perigee, when velocity is higher. Some burns in the EM-1 and EM-2 mission plans are much longer than burns executed with the Apollo and Space Shuttle vehicles. Burn length, as well as hyperbolic trajectories, has driven the use of the Encke-Beta numerical predictor by the predictor/corrector guidance algorithm in place of legacy analytic thrust and gravity integrals.
Robinson, Shane↗
A Runge-Kutta discontinuous finite element method for high speed flows
A Runge-Kutta discontinuous finite element method is developed for hyperbolic systems of conservation laws in two space variables. The discontinuous Galerkin spatial approximation to the conservation laws results in a system of ordinary differential equations which are marched in time using Runge-Kutta methods. Numerical results for the two-dimensional Burger's equation show that the method is (p+1)-order accurate in time and space, where p is the degree of the polynomial approximation of the solution within an element and is capable of capturing shocks over a single element without oscillations. Results for this problem also show that the accuracy of the solution in smooth regions is unaffected by the local projection and that the accuracy in smooth regions increases as p increases. Numerical results for the Euler equations show that the method captures shocks without oscillations and with higher resolution than a first-order scheme.
Bey, Kim S.↗
Automation of Overset Structured Mesh Generation onBoundary Representation Geometries
A scheme is presented for the automatic generation of structured overset meshes ongeometries that are defined by Boundary Representation (BRep) solids. The surface meshsystem consists of face, edge and node meshes corresponding to the three respective basicBRep entity types. A cut-cell method is introduced to improve robustness of the on-geometrydetermination test for a face mesh grid point. A geometric component tagging scheme is utilizedto enhance local grid point distribution on a configuration with a large range of geometricscales. A cap grid topology is automatically utilized around the trailing edge of wing and tailtips to enhance mesh quality and to enable more effective surface coverage. Robustness of thehyperbolic surface marching method is improved by replacing the point projection scheme witha walking scheme. Relaxation of surface grid spacing at concave corners enables automatedhigh quality hyperbolic volume mesh creation. Domain connectivity is automatically performedon the surface mesh system. A variety of test cases are presented including a re-entry capsule,two models of the Juncture Flow Experiment wing-body, five rotorcraft concept vehicles, andvarious components of the High-Lift Common Research Model from the High-Lift PredictionWorkshop 4.
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