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NASA NTRS1987

Green's function solution to heat transfer of a transparent gas through a tube

A heat transfer analysis of a transparent gas flowing through a circular tube of finite thickness is presented. This study includes the effects of wall conduction, internal radiative exchange, and convective heat transfer. The natural mathematical formulation produces a nonlinear, integrodifferential equation governing the wall temperature and an ordinary differential equation describing the gas temperature. This investigation proposes to convert the original system of equations into an equivalent system of integral equations. The Green's function method permits the conversion of an integrodifferential equation into a pure integral equation. The proposed integral formulation and subsequent computational procedure are shown to be stable and accurate.

Frankel, J. I.↗

NASA NTRS1994

Hydrodynamics of rotating stars and close binary interactions: Compressible ellipsoid models

We develop a new formalism to study the dynamics of fluid polytropes in three dimensions. The stars are modeled as compressible ellipsoids, and the hydrodynamic equations are reduced to a set of ordinary differential equations for the evolution of the principal axes and other global quantities. Both viscous dissipation and the gravitational radiation reaction are incorporated. We establish the validity of our approximations and demonstrate the simplicity and power of the method by rederiving a number of known results concerning the stability and dynamical oscillations of rapidly rotating polytropes. In particular, we present a generalization to compressible fluids of Chandrasekhar's classical results for the secular and dynamical instabilities of incompressible Maclaurin spheroids. We also present several applications of our method to astrophysical problems of great current interest, such as the tidal disruption of a star by a massive black hole, the coalescence of compact binaries driven by the emission of gravitational waves, and the development of instabilities in close binary systems.

Lai, Dong↗

NASA NTRS2003

Computation of Large-Scale Structure Jet Noise Sources With Weak Nonlinear Effects Using Linear Euler

An approximate technique is presented for the prediction of the large-scale turbulent structure sound source in a supersonic jet. A linearized Euler equations code is used to solve for the flow disturbances within and near a jet with a given mean flow. Assuming a normal mode composition for the wave-like disturbances, the linear radial profiles are used in an integration of the Navier-Stokes equations. This results in a set of ordinary differential equations representing the weakly nonlinear self-interactions of the modes along with their interaction with the mean flow. Solutions are then used to correct the amplitude of the disturbances that represent the source of large-scale turbulent structure sound in the jet.

Dahl, Milo D.↗

NASA NTRS2006

Magnetic Field Solver

The Magnetic Field Solver computer program calculates the magnetic field generated by a group of collinear, cylindrical axisymmetric electromagnet coils. Given the current flowing in, and the number of turns, axial position, and axial and radial dimensions of each coil, the program calculates matrix coefficients for a finite-difference system of equations that approximates a two-dimensional partial differential equation for the magnetic potential contributed by the coil. The program iteratively solves these finite-difference equations by use of the modified incomplete Cholesky preconditioned-conjugate-gradient method. The total magnetic potential as a function of axial (z) and radial (r) position is then calculated as a sum of the magnetic potentials of the individual coils, using a high-accuracy interpolation scheme. Then the r and z components of the magnetic field as functions of r and z are calculated from the total magnetic potential by use of a high-accuracy finite-difference scheme. Notably, for the finite-difference calculations, the program generates nonuniform two-dimensional computational meshes from nonuniform one-dimensional meshes. Each mesh is generated in such a way as to minimize the numerical error for a benchmark one-dimensional magnetostatic problem.

Ilin, Andrew V.↗

DOE OSTI2021

A scalable exponential-DG approach for nonlinear conservation laws: With application to Burger and Euler equations

In this work, we propose an Exponential DG framework for partial differential equations. We decompose 7 governing equations into linear and nonlinear parts to which we apply the discontinuous Galerkin 8 (DG) spatial discretization. In particular, we construct the linear part using Jacobian that effectively 9 capture stiff characteristics in the system. The former is integrated analytically, whereas the latter 10 is approximated. This approach i) is stable with a large Courant number (Cr > 1); ii) supports 11 high-order solutions both in time and space; iii) is computationally favorable compared to IMEX 12 DG methods with no preconditioner; iv) becomes comparable to explicit RKDG methods on uniform 13 mesh and beneficial on non-uniform grid for Euler equations; v) is scalable in a modern massively 14 parallel computing architecture due to its explicit nature of exponential time integrators and com15 pact communication stencil of DG method. Numerical results demonstrate the performance of our 16 proposed methods through various examples. We also discuss the stability and convergence analysis 17 for our exponential DG scheme in the context of Burgers equation.

42 ENGINEERING↗

NASA NTRS1990

An equation for the evolution of solar and stellar flare loops

An ordinary differential equation describing the evolution of a coronal loop subjected to a spatially uniform but time-varying heating rate is discussed. It is assumed that the duration of heating is long compared to the sound transit time through the loop, which is assumed to have uniform cross section area. The form of the equation changes as the loop evolves through three states: 'strong evaporation', 'scaling law behavior', and 'strong condensation'. Solutions to the equation may be used to compute the time dependence of the average coronal temperature and emission measure for an assumed temporal variation of the flare heating rate. The results computed from the model agree reasonably well with recent published numerical simulations and may be obtained with far less computational effort. The model is then used to study the May 21, 1980, solar flare observed by SMM and the giant April 12, 1985, flare observed on the star AD Leo.

Fisher, George H.↗

DOE OSTI2020

Variational nonlinear WKB in the Eulerian frame

Nonlinear WKB is a multiscale technique for studying locally plane-wave solutions of nonlinear partial differential equations (PDEs). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the generalized Lagrangian mean. On the other hand, the variational structure of nonlinear WKB in the conventional Eulerian frame remains mysterious. By exhibiting a variational principle for the extended equations from step (1) above, here we demonstrate that nonlinear WKB in the Eulerian frame is in fact variational. Remarkably, the variational principle for the extended system admits loops of relabeling transformations as a symmetry group. Noether’s theorem therefore implies that the extended Eulerian equations possess a family of circulation invariants parameterized by S 1 . As an illustrative example, we use our results to systematically deduce a variational model of high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Linearizing the BPS equations with vector and tensor multiplets

We analyse the BPS equations of N = (1, 0) supergravity theory in six dimensions coupled to a vector and tensor multiplet. We show how these BPS equations can be reduced to a set of linear differential equations. This system is triangular in that each layer of equations, while linear, is quadratically sourced by the solutions of the previous layers. We examine several explicit examples and discuss the construction of new families of microstate geometries. We expect that the result presented here will open up new branches of superstrata in which the momentum is encoded in a new class of charge carriers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Operator evolution from the similarity renormalization group and the Magnus expansion

The Magnus expansion is an efficient alternative to solving similarity renormalization group (SRG) flow equations with high-order, memory-intensive ordinary differential equation solvers. The numerical simplifications it offers for operator evolution are particularly valuable for in-medium SRG calculations, though challenges remain for difficult problems involving intruder states. Here we test the Magnus approach in an analogous but more accessible situation, which is the free-space SRG treatment of the spurious bound states arising from a leading-order chiral effective field theory (EFT) potential with very high cutoffs. In this work, we show that the Magnus expansion passes these tests and then use the investigations as a springboard to address various aspects of operator evolution that have renewed relevance in the context of the scale and scheme dependence of nuclear processes. These aspects include SRG operator flow with band- versus block-diagonal generators, universality for chiral EFT Hamiltonians and associated operators with different regularization schemes, and the impact of factorization arising from scale separation. Implications for short-range correlations physics and the possibilities for reconciling high- and low-resolution treatments of nuclear structure and reactions are discussed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Simulating Open Quantum Systems Using Hamiltonian Simulations

We present a novel method to simulate the Lindblad equation, drawing on the relationship between Lindblad dynamics, stochastic differential equations, and Hamiltonian simulations. We derive a sequence of unitary dynamics in an enlarged Hilbert space that can approximate the Lindblad dynamics up to an arbitrarily high order. This unitary representation can then be simulated using a quantum circuit that involves only Hamiltonian simulation and tracing out the ancilla qubits. There is no need for additional postselection in measurement outcomes, ensuring a success probability of one at each stage. Our method can be directly generalized to the time-dependent setting. We provide numerical examples that simulate both time-independent and time-dependent Lindbladian dynamics with accuracy up to the third order. Published by the American Physical Society 2024

Ding, Zhiyan (ORCID:000000018863403X)↗

DOE OSTI2021

iharm3D: Vectorized General Relativistic Magnetohydrodynamics

iharm3D is an open-source C code for simulating black hole accretion systems in arbitrary stationary spacetimes using ideal general-relativistic magnetohydrodynamics (GRMHD). It is an implementation of the HARM (“High Accuracy Relativistic Magnetohydrodynamics”) algorithm outlined in Gammie et al. (2003) with updates as outlined in McKinney & Gammie (2004) and Noble et al. (2006). The code is most directly derived from Ryan et al. (2015) but with radiative transfer portions removed. HARM is a conservative finite-volume scheme for solving the equations of ideal GRMHD, a hyperbolic system of partial differential equations, on a logically Cartesian mesh in arbitrary coordinates.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Parallel Algebraic Multigrid for Fusion and Higher-Order PDEs

Multigrid methods play a key role in large-scale scientific simulation because they are among the fastest and most scalable approaches for solving the underlying sparse linear systems of equations that arise from a wide array of Partial Differential Equation (PDE) discretizations. Algebraic multigrid (AMG) is a special type of multigrid method that depends only on the description of the linear system, giving it better portability and broader applicability than geometric multigrid, as it requires no explicit knowledge of the problem geometry. Even though these methods are widely used today, there are still applications where further development is needed. In this report, we focus on PDEs with higher-order terms (e.g., fourth order), concentrating on a PDE that arises in tokamak edge plasma simulations (a tokamak is a machine that confines a plasma using magnetic fields and is believed to be the leading plasma confinement concept for future fusion power plants). General multigrid relaxes a linear system on coarser grids and reverses this process with interpolation, but standard AMG methods struggle with the aforementioned higher-order PDEs. We investigate cyclic coarsening and interpolation heuristics, as well as new iterative approximation methods of refining the solution at each grid to improve the existing multigrid approach. To this end, we ensure that these techniques are transferable to a parallelized setting with LLNL’s supercomputers.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1967

Periodic solutions of a class of hyperbolic equations containing a small parameter.

Periodic solutions of hyperbolic equations containing small parameter by extending Cesari method for differential equations

PERIODIC FUNCTION↗

NASA NTRS1971

Inertial navigation systems analysis.

This volume offers the avionic systems engineer a fundamental exposition of the mechanization and error analysis of inertial navigation systems. While the material is applicable to spacecraft and undersea navigation, emphasis is placed upon terrestrial applications on or slightly above the earth's surface. As a result, practical considerations are geared toward those aircraft navigation systems of particular current interest. Extensive use is made of perturbation techniques to develop linearized system equations, whose solutions closely approximate those obtained by nonlinear differential equations. A unified error analysis technique is developed that is applicable to virtually all system configurations. The technique provides a greatly simplified method for comparing the performance of competing system configurations.

Britting, K. R.↗

NASA NTRS1972

A numerical investigation of film cooling as an alternate thermal protection system for spacecraft during atmospheric reentry

Research was undertaken to determine the mass required to protect a surface during atmospheric reentry by injecting a cold gas tangentially along the surface. A consideration of this problem revealed that the boundary layer equations could be employed to approximate the most important flow configurations which occur in film cooling. However, even with this simplification, the equations to be solved are complex, nonlinear, coupled partial differential equations. Due to the relative weakness of classical methods of boundary layer analysis, finite difference methods were chosen as the best approach to the formidable problem under consideration. Instead of blindly adopting a proposed technique and trying to force it to the problem, a more systematic study was deemed to be not only desirable but necessary. A relatively basic study of the application of finite difference methods was undertaken with the objective of developing an algorithm capable of obtaining accurate solutions to film cooling problems.

Clausing, A. M.↗

NASA NTRS1973

Marshall system for aerospace system simulation (MARSYAS), user's manual

The capabilities of the Marshall system for aerospace system simulation (MARSYAS) and how to use it are described. MARSYAS is a software system that allows easy setup and control of the simulation of the dynamics of large physical systems on a digital computer. The physical systems are modeled in the form of block diagrams or equations. The blocks can have multiple inputs and multiple outputs, and they can be nested to form hierarchies. The block diagrams can contain transfer functions, nonlinear and logical functions, equations, analog computer elements and FORTRAN programs. The input format of the equations can be combinations of nonlinear, time-varying differential equations and algebraic equations in their original format. MARSYAS could also serve as a storage and retrieval system for models as a basis for a model configuration control system on a central time-shared computer. The outputs of the simulation system can be not only time-responses but also other analysis data such as frequency response, power spectrum and stability parameters. The MARSYAS translator is written in FORTRAN running on the Univac 1108 computer under the EXEC 8 operating system.

Ventre, A.↗

NASA NTRS1974

Fast direct numerical solution of the nonhomogeneous Cauchy-Riemann equations

A fast direct (noniterative) 'Cauchy-Riemann Solver' is developed for solving the finite-difference equations representing systems of first-order elliptic partial differential equations in the form of the nonhomogeneous Cauchy-Riemann equations. The method is second-order accurate and requires approximately the same computer time as a fast cyclic-reduction Poisson solver. The accuracy and efficiency of the direct solver are demonstrated in an application to solving an example problem in aerodynamics: subsonic inviscid flow over a biconvex airfoil. The analytical small-perturbation solution contains singularities, which are captured well by the computational technique. The algorithm is expected to be useful in nonlinear subsonic and transonic aerodynamics.

Lomax, H.↗