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At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1989

Galerkin/Runge-Kutta discretizations for parabolic equations with time-dependent coefficients

A new class of fully discrete Galerkin/Runge-Kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients. Unlike any classical counterpart, this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction. In support of this claim, error estimates are proved, and computational results are presented. Additionally, since the time stepping equations involve coefficient matrices changing at each time step, a preconditioned iterative technique is used to solve the linear systems only approximately. Nevertheless, the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients. Furthermore, it is noted that special Runge-Kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method.

Keeling, Stephen L.↗
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 NTRS2004

Adaptive Numerical Dissipative Control in High Order Schemes for Multi-D Non-Ideal MHD

The goal is to extend our adaptive numerical dissipation control in high order filter schemes and our new divergence-free methods for ideal MHD to non-ideal MHD that include viscosity and resistivity. The key idea consists of automatic detection of different flow features as distinct sensors to signal the appropriate type and amount of numerical dissipation/filter where needed and leave the rest of the region free of numerical dissipation contamination. These scheme-independent detectors are capable of distinguishing shocks/shears, flame sheets, turbulent fluctuations and spurious high-frequency oscillations. The detection algorithm is based on an artificial compression method (ACM) (for shocks/shears), and redundant multi-resolution wavelets (WAV) (for the above types of flow feature). These filter approaches also provide a natural and efficient way for the minimization of Div(B) numerical error. The filter scheme consists of spatially sixth order or higher non-dissipative spatial difference operators as the base scheme for the inviscid flux derivatives. If necessary, a small amount of high order linear dissipation is used to remove spurious high frequency oscillations. For example, an eighth-order centered linear dissipation (AD8) might be included in conjunction with a spatially sixth-order base scheme. The inviscid difference operator is applied twice for the viscous flux derivatives. After the completion of a full time step of the base scheme step, the solution is adaptively filtered by the product of a 'flow detector' and the 'nonlinear dissipative portion' of a high-resolution shock-capturing scheme. In addition, the scheme independent wavelet flow detector can be used in conjunction with spatially compact, spectral or spectral element type of base schemes. The ACM and wavelet filter schemes using the dissipative portion of a second-order shock-capturing scheme with sixth-order spatial central base scheme for both the inviscid and viscous MHD flux derivatives and a fourth-order Runge-Kutta method are denoted.

Yee, H. C.↗
NASA NTRS1995

A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations

A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.

Liu, Yen↗
DOE OSTI2020

Modelling of nanometer scale dust grains in tokamak

Dust poses a serious threat to tokamak operation and safety. It is important to study the behaviour of dust grains under tokamak's discharge conditions, which depends heavily on their size and charge. Existing simulations mainly address issues on dust grains with radii larger than 1 μm, in which case, the drift effect due to electromagnetic fields can be safely ignored. For nanometer scale dust grains, however, the drift effect becomes significant and a new model based on guiding-centre system needs to be established. In this work, the NDS has been done under BOUT++ framework. The simulation contains two parts. Part one, NDS evaluates the charging and ablation processes of the dust grains. In the second part, the guiding-centre orbits of dust particles are tracked in tokamak plasmas, whose parameters are obtained from BOUT++, a highly desirable C++ code package for performing parallel plasma fluid simulations with an arbitrary number of equations in 3D curvilinear coordinates. The orbit of nanodust dynamics is described by guiding centre equations for simplicity, and these equations are numerically solved by conventional fourth-order Runge Kutta method. Simulations provide results such as trajectories and evolutions of dust particles with different sizes and velocities for different tokamak geometries. Results show tungsten dust grains with a radius of a few nanometers launched from outer midplane will oscillate before totally ablated in C-Mod. The oscillation in this case is driven by the ion drag force. Larger Nanodust with a radius of 100 nm, on the contrary, cannot be completely constrained by the electromagnetic field. In conclusion, the high plasma temperature and density in the seperatrix region causes severe dust ablation, resulting in total ablation within several ms.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Anisotropic fluid dynamical simulations of heavy-ion collisions

We present VAH, a (3+1)–dimensional simulation that evolves the far-from-equilibrium quark-gluon plasma produced in ultrarelativistic heavy-ion collisions with anisotropic fluid dynamics. We solve the hydrodynamic equations on an Eulerian grid using the Kurganov–Tadmor algorithm in combination with a new adaptive Runge–Kutta method. Our numerical scheme allows us to start the simulation soon after the nuclear collision, largely avoiding the need to integrate it with a separate pre-equilibrium dynamics module. We test the code's performance by simulating on the Eulerian grid conformal and non-conformal Bjorken flow as well as conformal Gubser flow, whose (0+1)–dimensional solutions are precisely known. Finally, we compare non-conformal anisotropic hydrodynamics to second-order viscous hydrodynamics in central Pb+Pb collisions and find that the former's longitudinal flow profile responds more consistently to the fluid's gradients along the spacetime rapidity direction.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Storm Surge Modeling as an Application of Local Time-Stepping in MPAS-Ocean

This paper presents the first practical application of local time-stepping (LTS) schemes in the Model for Prediction Across Scales-Ocean (MPAS-O). We use LTS schemes in a single-layer, global ocean model that predicts the storm surge around the eastern coast of the United States during Hurricane Sandy. The variable-resolution meshes used are of unprecedentedly high resolution in MPAS-O, containing cells as small as 125 m wide in Delaware Bay. It is shown that a particular, third-order LTS scheme (LTS3) produces sea-surface height solutions that are of comparable quality to solutions produced by the classical four-stage, fourth-order Runge-Kutta method (RK4) with a uniform time step on the same meshes. Furthermore, LTS3 is up to 35% faster in the best cases considered, where the number of cells using the coarse time-step relative to those using the fine time-step is as low as 1:1. This shows that LTS schemes are viable for use in MPAS-O with the added benefit of substantially less computational cost. The results of these performance experiments inform us of the requirements for efficient mesh design and configuration of LTS regions for LTS schemes. In particular, we see that for LTS to be efficient on a given mesh, it is important to have enough cells using the coarse time-step relative to those using the fine time-step, typically at least 1:5 to see an increase in performance.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Clebsch canonization of Lie–Poisson systems

We propose a systematic procedure called the Clebsch canonization for obtaining a canonical Hamiltonian system that is related to a given Lie–Poisson equation via a momentum map. We describe both coordinate and geometric versions of the procedure, the latter apparently for the first time. We also find another momentum map so that the pair of momentum maps constitute a dual pair under a certain condition. The dual pair gives a concrete realization of what is commonly referred to as collectivization of Lie–Poisson systems. It also implies that solving the canonized system by symplectic Runge–Kutta methods yields so-called collective Lie–Poisson integrators that preserve the coadjoint orbits and hence the Casimirs exactly. We give a couple of examples, including the Kida vortex and the heavy top on a movable base with controls, which are Lie–Poisson systems on $\mathfrak{so}$ (2,1)* and ($\mathfrak{se}$ (3) ⋉ $\mathbb{R}^3$)* and, respectively.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1971

Computer program for analysis of imperfection sensitivity of ring stiffened shells of revolution

A FORTRAN 4 digital computer program is presented for the initial postbuckling and imperfection sensitivity analysis of bifurcation buckling modes for ring-stiffened orthotropic multilayered shells of revolution. The boundary value problem for the second-order contribution to the buckled state was solved by the forward integration technique using the Runge-Kutta method. The effects of nonlinear prebuckling states and live pressure loadings are included.

Cohen, G. A.↗
NASA NTRS1972

Analytical treatment of gas flows through multilayer insulation, project 1

A theoretical investigation of gas flow inside a multilayer insulation system was made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was also investigated. It was shown that when the relaxation time is less than the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparisons with experimental data were made.

Lin, J. T.↗
NASA NTRS1973

Tables for Supersonic Flow of Helium Around Right Circular Cones at Zero Angle of Attack

The results of the calculation of supersonic flow of helium about right circular cones at zero angle of attack are presented in tabular form. The calculations were performed using the Taylor-Maccoll theory. Numerical integrations were performed using a Runge-Kutta method for second-order differential equations. Results were obtained for cone angles from 2.5 to 30 degrees in regular increments of 2.5 degrees. In all calculations the desired free-stream Mach number was obtained to five or more significant figures.

Sims, J. L.↗
NASA NTRS1973

Analysis of gas flow through a multilayer insulation system.

A theoretical investigation of gas flow inside a multilayer insulation system has been made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas mixture was obtained by considering the diffusion mechanism of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was investigated. It has been shown that when the relaxation time is small compared with the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given and comparison with experimental data was made.

Lin, J. T.↗
NASA NTRS1973

Thermal engineering research

A theoretical investigation of gas flow inside a multilayer insulation system has been made for the case of the broadside pumping process. A set of simultaneous first-order differential equations for the temperature and pressure of the gas mixture was obtained by considering the diffusion mechanism of the gas molecules through the perforations on the insulation layers. A modified Runge-Kutta method was used for numerical experiment. The numerical stability problem was investigated. It has been shown that when the relaxation time is small compared with the time period over which the gas properties change appreciably, the set of differential equations can be replaced by a set of algebraic equations for solution. Numerical examples were given, and comparisons with experimental data were made.

Shih, C. C.↗
NASA NTRS1977

An optimal Space Shuttle ascent trajectory for the first orbital flight test

An optimal solution of the ascent trajectory of the Space Shuttle for the first orbital flight test is presented; the optimization is a minimum propellant, four-control problem in yaw angle, roll angle, pitch angle and vacuum thrust of each Space Shuttle main engine. Piecewise linear segments with juncture points treated as parameters are employed to model the controls. Equations of motion for a three-dimensional flight with pitch plane moment balance about an oblate are integrated numerically with a fourth-order Runge-Kutta method; two- and one-dimensional cubic spline function curve fits of aerodynamic coefficients are used during the first and second stages, respectively. The constraint minimization problem is solved with the Davidon-Fletcher-Powell function method.

Johnson, I. L., Jr.↗
NASA NTRS1978

Non-linear effects on sound propagation in a variable area duct carrying high subsonic Mach number flows

The general one-dimensional non-linear equation for the acoustic velocity potential in a variable area duct carrying high subsonic Mach number flows is presented and solved numerically using an implicit finite difference scheme. For the linearized equation with no flow, the present scheme is compared with the exact solution and the fourth-order Runge-Kutta method with excellent agreement for dimensionless time periods. Non-linear solutions are more sensitive to Mach numbers and exciting amplitudes and less sensitive to exciting frequencies. In general, non-linear effects can be safely neglected for low Mach number flows but must be accounted for when high Mach number flows are encountered.

Lumsdaine, E.↗
NASA NTRS1981

Elasto visco-plastic flow with special attention to boundary conditions

A simple but nontrivial steady-state creeping elasto visco-plastic (Maxwell fluid) radial flow problem is analyzed, with special attention given to the effects of the boundary conditions. Solutions are obtained through integration of a governing equation on stress using the Runge-Kutta method for initial value problems and finite differences for boundary value problems. A more general approach through the finite element method, an approach that solves for the velocity field rather than the stress field and that is applicable to a wide range of problems, is presented and tested using the radial flow example. It is found that steady-state flows of elasto visco-plastic materials are strongly influenced by the state of stress of material as it enters the region of interest. The importance of this boundary or initial condition in analyses involving materials coming into control volumes from unusual stress environments is emphasized.

Shimazaki, Y.↗