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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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NASA NTRS1984

Orbital stability of the unseen solar companion linked to periodic extinction events

Evidence from three-dimensional numerical modelling is presented that only cometary orbits with a limited range in inclination with respect to the galactic plane are formally stable for the length of time required to cause periodic extinction events. The calculations were done using Cowell's method employing a fourth-order Runge-Kutta integration scheme in an inertial reference frame in orbit about the Galaxy. Tidal perturbations in the radial direction due to the Galaxy and the Coriolis forces are included. The vertical component of the gravitational field of the galactic disk is superimposed on these forces. The results indicate that orbits for Nemesis that are inclined at more than 30 deg to the galactic plane are not allowed and suggests that the search for Nemesis should be concentrated toward the plane of the Galaxy. Perturbations by passing stars or molecular clouds may make even the low-inclination orbits unstable.

Torbett, M. V.↗

NASA NTRS1982

Computation of transonic viscous-inviscid interacting flow

Transonic viscous-inviscid interaction is considered using the Euler and inverse compressible turbulent boundary-layer equations. Certain improvements in the inverse boundary-layer method are mentioned, along with experiences in using various Runge-Kutta schemes to solve the Euler equations. Numerical conditions imposed on the Euler equations at a surface for viscous-inviscid interaction using the method of equivalent sources are developed, and numerical solutions are presented and compared with experimental data to illustrate essential points.

Whitfield, D. L.↗

NASA NTRS1983

Computation of transonic viscous-inviscid interacting flow

Whitfield, D. L.↗

NASA NTRS1967

Approximate solutions of Runge-Kutta equations

APPROXIMATION METHOD↗

NASA NTRS1978

A method of weighted residuals with trigonometric basis functions for sound transmission in circular ducts

The method of weighted residuals (MWR) in the form of a modified Galerkin method with trigonometric basis functions is used to compute the transmission of sound in an axisymmetric duct. The method is used to generate the axial wave number for uniform ducts. These are compared with exact solutions generated by a formal eigenvalue routine in the hard-wall case and a Runge-Kutta integration eigenvalue scheme in the soft-wall case. The method is applicable to both flow and no-flow cases.

Vo, P. T.↗

NASA NTRS1985

DEAN: A program for dynamic engine analysis

The Dynamic Engine Analysis program, DEAN, is a FORTRAN code implemented on the IBM/370 mainframe at NASA Lewis Research Center for digital simulation of turbofan engine dynamics. DEAN is an interactive program which allows the user to simulate engine subsystems as well as a full engine systems with relative ease. The nonlinear first order ordinary differential equations which define the engine model may be solved by one of four integration schemes, a second order Runge-Kutta, a fourth order Runge-Kutta, an Adams Predictor-Corrector, or Gear's method for still systems. The numerical data generated by the model equations are displayed at specified intervals between which the user may choose to modify various parameters affecting the model equations and transient execution. Following the transient run, versatile graphics capabilities allow close examination of the data. DEAN's modeling procedure and capabilities are demonstrated by generating a model of simple compressor rig.

Sadler, G. G.↗

NASA NTRS1985

DEAN - A program for Dynamic Engine ANalysis

The Dynamic Engine Analysis Program, DEAN, is a FORTRAN code implemented on the IBM/370 mainframe at NASA Lewis Research Center for digital simulation of turbofan engine dynamics. DEAN is an interactive program which allows the user to simulate engine subsystems as well as full engine systems with relative ease. The nonlinear first order ordinary differential equations which define the engine model may be solved by one of four integration schemes, a second order Runge-Kutta, a fourth order Runge-Kutta, an Adams Predictor-Corrector, or Gear's method for still systems. The numerical data generated by the model equations are displayed a specified intervals between which the user may choose to modify various parameters affecting the model equations and transient execution. Following the transient run, versatile graphics capabilities allow close examination of the data. DEAN's modeling procedure and capabilities are demonstrated by generating a model of simple compressor rig.

Sadler, G. G.↗

NASA NTRS1966

Solutions of differential equations by evaluations of functions.

Runge-Kutta fourth order formula used to obtain approximate solution to differential equations

RUNGE-KUTTA INTEGRATION↗

NASA NTRS1977

An ILLIAC program for the numerical simulation of homogeneous incompressible turbulence

An algorithm and ILLIAC computer program, developed for the simulation of homogeneous incompressible turbulence in the presence of an applied mean strain, are described. The turbulence field is represented spatially by a truncated triple Fourier series (spectral method) and followed in time using a fourth-order Runge-Kutta algorithm. These include: (1) transformation of variables suggested by Taylor's sudden-distortion theory; (2) implicit viscous diffusion by use of an integrating factor; (3) implicit pressure calculation suggested by Taylor's sudden-distortion theory, and (4) inexpensive control of aliasing by random and phased coordinate shifts.

Rogallo, R. S.↗

DOE OSTI2025

Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with variable coefficients

Here, we consider the adaptive-rank integration of multi-dimensional time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with high-order diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with a projection-based adaptive-rank algorithm structured around two key strategies: (i) constructing dimension-wise subspaces using a novel atypical extended Krylov strategy, and (ii) efficiently solving the basis coefficient matrix with a preconditioned GMRES solver. The low-rank decomposition is performed in 2D using SVD and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format. For d-dimensional problems (here, d = 2 or 3), the computational complexity and memory storage of the approach are found numerically to scale as and $\mathscr{O}(Nr^2) + \mathscr{O} (r^{d+1})$ and $\mathscr{O}(Nr) + \mathscr{O} (r^{d})$, respectively, with the one-dimensional resolution and the maximal rank during the Krylov iteration (which we find to be largely independent of on our numerical examples). We present numerical examples that illustrate the advertised properties of the algorithm.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1979

Nonlinear wave choked inlets

The quasi-one dimensional flow program was modified in two ways. The Runge-Kutta subroutine was replaced with a subroutine which used a modified divided difference form of the Adams Pece method and the matrix inversion routine was replaced with a pseudo inverse routine. Calculations were run using both the original and modified programs. Comparison of the calculations showed that the original Runge-Kutta routine could not detect singularity near the throat and was integrating across it. The modified version was able to detect the singularity and therefore gave more valid calculations.

Source record↗

NASA NTRS1984

On the solution of problems involving impact type loading

The response of beam and plate members to pulse and impact loading is investigated, and numerical methods are analyzed and compared. The equations of motion, the finite difference and finite element methods, the time integration/Runge-Kutta techniques, and material modeling are discussed in detail. It is found that both the finite difference and the finite element methods could be accurately employed to discretize the spatial variation in the displacements. Central differences or a fourth order Runge-Kutta algorithm could be used for the time integration. The total energy of the system would give the stability and accuracy of the solution. Results showed that the finite element method provided better efficiency in obtaining accurate solution than the finite difference method when a scalar processor is used. However, the finite difference method was more efficient on the vector processor. Therefore, the most efficient method of solution depends on the type of computer present for the analysis.

Moyer, E. T., Jr.↗

NASA NTRS1985

Numerical solution of the Euler equations for high-speed, blended wing-body configurations

Simulation of high-speed three-dimensional flow about blended wing-body combinations is investigated. A finite-volume explicit scheme with Runge-Kutta time integration is used to solve the compressible Euler equations in order to simulate the flow. The method, augmented by carefully chosen dissipative terms and convergence accelerators such as enthalpy damping and maximum local time-stepping, has been found to be very efficient in solving high-speed flows involving strong shocks. An analytic method for generating the body geometry and an algebraic method for quasi-three-dimensional grid generation are described. Results are presented for various blended wing-body configurations at different Mach numbers and angles of attack.

Moitra, A.↗

NASA NTRS1974

Some self starting integrators for x Prime equals f (x, t)

The integration is discussed of the vector differential equation X = F(x, t) from time t sub i to t sub (i = 1) where only the values of x sub i are available for the the integration. No previous values of x or x prime are used. Using an orbit integration problem, comparisons are made between Taylor series integrators and various types and orders of Runge-Kutta integrators. A fourth order Runge-Kutta type integrator for orbital work is presented, and approximate (there may be no exact) fifth order Runge-Kutta integrators are discussed. Also discussed and compared is a self starting integrator ising delta f/delta x. A numerical method for controlling the accuracy of integration is given, and the special equations for accurately integrating accelerometer data are shown.

Lear, W. M.↗

NASA NTRS1985

SSME Turbopump Turbine Computations

A two-dimensional viscous code was developed to be used in the prediction of the flow in the SSME high-pressure turbopump blade passages. The rotor viscous code (RVC) employs a four-step Runge-Kutta scheme to solve the two-dimensional, thin-layer Navier-Stokes equations. The Baldwin-Lomax eddy-viscosity model is used for these turbulent flow calculations. A viable method was developed to use the relative exit conditions from an upstream blade row as the inlet conditions to the next blade row. The blade loading diagrams are compared with the meridional values obtained from an in-house quasithree-dimensional inviscid code. Periodic boundary conditions are imposed on a body-fitted C-grid computed by using the GRAPE GRids about Airfoils using Poisson's Equation (GRAPE) code. Total pressure, total temperature, and flow angle are specified at the inlet. The upstream-running Riemann invariant is extrapolated from the interior. Static pressure is specified at the exit such that mass flow is conserved from blade row to blade row, and the conservative variables are extrapolated from the interior. For viscous flows the noslip condition is imposed at the wall. The normal momentum equation gives the pressure at the wall. The density at the wall is obtained from the wall total temperature.

Jorgenson, P. G. E.↗

NASA NTRS1985

Convergence acceleration of iterative solutions of Euler equations for transonic flow computations

In this paper, two acceleration techniques for Euler calculations are investigated. The first technique is an extrapolation procedure based on the Power Method; it is applicable when the iterative matrix has dominant eigenvalues. Both real and complex conjugate roots are allowed. The second technique is a generalization of the Minimal Residual Method, where the extrapolation step consists of a weighted combination of the corrections at different iteration levels and the weights are chosen to minimize the Lz norm of the residual. Numerical results, using Jameson's Runge-Kutta Multigrid Code, are presented. The extra computational work to apply either technique is negligible and the extra storage is not a problem on current supercomputers.

Hafez, M.↗

DOE OSTI2025

High order interpolation of magnetic fields with vector potential reconstruction for particle simulations

We propose a method for interpolating divergence-free continuous magnetic fields via vector potential reconstruction using Hermite interpolation, which ensures high-order continuity for applications requiring adaptive, high-order ordinary differential equation (ODE) integrators, such as the Dormand-Prince method. The method provides C(m) continuity and achieves high-order accuracy, making it particularly suited for particle trajectory integration and Poincaré section analysis under optimal integration order and timestep adjustments. Through numerical experiments, we demonstrate that the Hermite interpolation method preserves volume and continuity, which are critical for conserving toroidal canonical momentum and magnetic moment in guiding center simulations, especially over long-term trajectory integration. Furthermore, we analyze the impact of insufficient derivative continuity on Runge-Kutta schemes and show how it degrades accuracy at low error tolerances, introducing discontinuity-induced truncation errors. Lastly, we demonstrate performant Poincaré section analysis in two relevant settings of field data collocated from finite element meshes.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2024

A high-order explicit Runge-Kutta approximation technique for the shallow water equations

Here, we introduce a high-order space–time approximation of the Shallow Water Equations with sources that is invariant-domain preserving (IDP), well-balanced with respect to rest states, and employs a novel explicit Runge–Kutta (ERK) introduced in Ern and Guermond (SIAM J. Sci. Comput. 44(5), A3366–A3392, 2022) for systems of non-linear conservation equations. The resulting method is then numerically illustrated through verification and validation.

97 MATHEMATICS AND COMPUTING↗