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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Achieving Higher Order Accuracy in Space in Hydrodynamic Simulations of Self-Gravitating Gas

Modern astrophysical simulation codes employ a variety of numerical algorithms capable of achieving higher-order accuracy in both space and time. Albeit they succeed in achieving an effective higher spatial resolution and in suppressing the numerical damping of waves, to our knowledge, all current astrophysical simulations invoking self-gravity are limited to second-order accuracy in space. If we can devise an algorithm to evaluate self-gravity with a higher-order spatial accuracy, we can better the evaluation of the gravitational acceleration and gravitational energy release which dictate the evolution of many astrophysical systems. Herein, we present a numerical algorithm for self-gravitating hydrodynamics capable of achieving fourth-order accuracy for a given density distribution on a Cartesian uniform grid. First, we derive the cell-averaged gravitational potential at fourth-order accuracy from the cell-averaged density by solving the Poisson equation. Next, we obtain the cell average of the product of the density and gravitational acceleration, which differs from the cell-averaged density multiplied by the cell-averaged gravitational acceleration. We then show the verification of the algorithm by applying it to critical test problems: (1) maintaining equilibria of self-gravitating slabs, even upon advection, (2) evolving a polytropic sphere with a massive power-law envelope, and (3) conservation of specific entropy during the propagation of a sound wave.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

Achieving Higher Order Accuracy in Space in Hydrodynamic Simulations of Self-Gravitating Gas

Modern astrophysical simulation codes employ a variety of numerical algorithms capable of achieving higher-order accuracy in both space and time. Albeit they succeed in achieving an effective higher spatial resolution and in suppressing the numerical damping of waves, to our knowledge, all current astrophysical simulations invoking self-gravity are limited to second-order accuracy in space. If we can devise an algorithm to evaluate self-gravity with a higher-order spatial accuracy, we can better the evaluation of the gravitational acceleration and gravitational energy release which dictate the evolution of many astrophysical systems. Herein, we present a numerical algorithm for self-gravitating hydrodynamics capable of achieving fourth order accuracy for a given density distribution on a Cartesian uniform grid. First, we derive the cell-averaged gravitational potential at fourth-order accuracy from the cell-averaged density by solving the Poisson equation. Next, we obtain the cell average of the product of the density and gravitational acceleration, which differs from the cell-averaged density multiplied by the cell-averaged gravitational acceleration. We then show the verification of the algorithm by applying it to critical test problems: (1) maintaining equilibria of self-gravitating slabs, even upon advection, (2) evolving a polytropic sphere with a massive power-law envelope, and (3) conservation of specific entropy during the propagation of a sound wave.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS1996

Artificial Boundary Conditions Based on the Difference Potentials Method

While numerically solving a problem initially formulated on an unbounded domain, one typically truncates this domain, which necessitates setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The issue of setting the ABC's appears to be most significant in many areas of scientific computing, for example, in problems originating from acoustics, electrodynamics, solid mechanics, and fluid dynamics. In particular, in computational fluid dynamics (where external problems present a wide class of practically important formulations) the proper treatment of external boundaries may have a profound impact on the overall quality and performance of numerical algorithms. Most of the currently used techniques for setting the ABC's can basically be classified into two groups. The methods from the first group (global ABC's) usually provide high accuracy and robustness of the numerical procedure but often appear to be fairly cumbersome and (computationally) expensive. The methods from the second group (local ABC's) are, as a rule, algorithmically simple, numerically cheap, and geometrically universal; however, they usually lack accuracy of computations. In this paper we first present a survey and provide a comparative assessment of different existing methods for constructing the ABC's. Then, we describe a relatively new ABC's technique of ours and review the corresponding results. This new technique, in our opinion, is currently one of the most promising in the field. It enables one to construct such ABC's that combine the advantages relevant to the two aforementioned classes of existing methods. Our approach is based on application of the difference potentials method attributable to V. S. Ryaben'kii. This approach allows us to obtain highly accurate ABC's in the form of certain (nonlocal) boundary operator equations. The operators involved are analogous to the pseudodifferential boundary projections first introduced by A. P. Calderon and then also studied by R. T. Seeley. The apparatus of the boundary pseudodifferential equations, which has formerly been used mostly in the qualitative theory of integral equations and PDE'S, is now effectively employed for developing numerical methods in the different fields of scientific computing.

Tsynkov, Semyon V.↗
NASA NTRS1973

Solution of quadratic matrix equations for free vibration analysis of structures.

An efficient digital computer procedure and the related numerical algorithm are presented herein for the solution of quadratic matrix equations associated with free vibration analysis of structures. Such a procedure enables accurate and economical analysis of natural frequencies and associated modes of discretized structures. The numerically stable algorithm is based on the Sturm sequence method, which fully exploits the banded form of associated stiffness and mass matrices. The related computer program written in FORTRAN V for the JPL UNIVAC 1108 computer proves to be substantially more accurate and economical than other existing procedures of such analysis. Numerical examples are presented for two structures - a cantilever beam and a semicircular arch.

Gupta, K. K.↗
DOE OSTI2020

Experimental and theoretical study of energy transfer in a chromophore triad: What makes modeling dynamics successful?

Simulation of electronic dynamics in realistically large molecular systems is a demanding task that has not yet achieved the same level of quantitative prediction already realized for its static counterpart. This is particularly true for processes occurring beyond the Born–Oppenheimer regime. Non-adiabatic molecular dynamics (NAMD) simulations suffer from two convoluted sources of error: numerical algorithms for dynamics and electronic structure calculations. While the former has gained increasing attention, particularly addressing the validity of ad hoc methodologies, the effect of the latter remains relatively unexplored. Indeed, the required accuracy for electronic structure calculations to reach quantitative agreement with experiment in dynamics may be even more strict than that required for static simulations. In this study, we address this issue by modeling the electronic energy transfer in a donor–acceptor–donor (D–A–D) molecular light harvesting system using fewest switches surface hopping NAMD simulations. In the studied system, time-resolved experimental measurements deliver complete information on spectra and energy transfer rates. Subsequent modeling shows that the calculated electronic transition energies are “sufficiently good” to reproduce experimental spectra but produce over an order of magnitude error in simulated dynamical rates. We further perform simulations using artificially shifted energy gaps to investigate the complex relationship between transition energies and modeled dynamics to understand factors affecting non-radiative relaxation and energy transfer rates.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Quantum Fourier transform revisited

Summary The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank‐1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix‐2 QFT decomposition to a radix‐ d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

Camps, Daan↗
DOE OSTI2020

Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. In this paper, we present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch angle scattering of a particle beam, and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long time accuracy of the proposed algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. We present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method.

Zhang, Xin↗
NASA NTRS2023

Implementation, Realization and an Effective Solver of Two-Equation Turbulence Models

Currently, when the Reynolds-Averaged Navier-Stokes (RANS) equations are solved using turbulence modeling, most often the one-equation model of Spalart and Allmaras is used. Then, it is only necessary to solve the RANS equations in conjunction with a single transport equation for modeling turbulence. For this model, considerable assessment and analysis has been performed, allowing the possibility of a reliable solution method for an eddy viscosity required to compute the Reynolds stresses in the RANS equations. Such evaluation along with analysis has not been achieved to realize similar performance with two-equation models of the k-w type. The primary objective of this paper is to present and discuss the components of an effective numerical algorithm for solving the RANS equations and the two transport equations of k-w type turbulence models. All the important details of the turbulence model as actually implemented are given, which is sometimes not done in various papers considering such modeling. The viability and effectiveness of this solution algorithm are demonstrated by solving both two-dimensional and three-dimensional aerodynamic flows. In all applications, a linear rate of convergence without oscillations or other evidence of unstable behavior is observed. This behavior is also particularly true when the proposed algorithm is applied to systematically renewed mesh sequences, which is generally not observed with algorithms solving more than one transport equation. Thus, numerical integration errors are systematically reduced, allowing for a significantly more reliable assessment of the effectiveness of the turbulence model. Additionally, in this paper, analysis of the solution algorithm, including linear stability, is also performed for a particular flow problem.

Turbulence↗
NASA NTRS1993

Improved Algorithm Computes Incompressible Flow

Algorithm numerically integrates Navier-Stokes equations of time-dependent or steady flow of incompressible, viscous fluid. Simulates realistic three-dimensional flows bounded by stationary or moving surface(s) of complicated shape(s). Only two arbitrary parameters specified by user. Algorithm described in "Numerical Simulation Of Flow Through An Artificial Heart" (ARC-12478). Algorithm tested by application to number of steady and unsteady flows, including flow in artificial heart discussed in noted prior article.

Rogers, Stuart E.↗
NASA NTRS1987

Numerical simulation of three-dimensional unsteady vortex flow using a compact vorticity-velocity algorithm

A numerical algorithm is presented which is used to solve the unsteady, fully three-dimensional, incompressible Navier-Stokes equations in vorticity-velocity variables. A discussion of the discrete approximation scheme is presented as well as the solution method used to solve the resulting algebraic set of difference equations. Second order spatial and temporal accuracy is verified through solution comparisons with exact results obtained for steady three-dimensional stagnation point flow and unsteady axisymmetric vortex spin-up. In addition, results are presented for the problem of unsteady bubble-type vortex breakdown with emphasis on internal bubble dynamics and structure.

Gatski, T. B.↗
NASA NTRS1991

Implementation of a partitioned algorithm for simulation of large CSI problems

The implementation of a partitioned numerical algorithm for determining the dynamic response of coupled structure/controller/estimator finite-dimensional systems is reviewed. The partitioned approach leads to a set of coupled first and second-order linear differential equations which are numerically integrated with extrapolation and implicit step methods. The present software implementation, ACSIS, utilizes parallel processing techniques at various levels to optimize performance on a shared-memory concurrent/vector processing system. A general procedure for the design of controller and filter gains is also implemented, which utilizes the vibration characteristics of the structure to be solved. Also presented are: example problems; a user's guide to the software; the procedures and algorithm scripts; a stability analysis for the algorithm; and the source code for the parallel implementation.

Alvin, Kenneth F.↗
DOE OSTI2021

Continuum gyrokinetic simulations of edge plasmas in single-null geometries

The first continuum gyrokinetic calculations of electrostatic ion scale turbulence are presented for the case of a diverted tokamak geometry. The simulation model solves the long-wavelength limit of the full-F gyrokinetic equation for ion species coupled to the quasi-neutrality equation for electrostatic potential variations, where a fluid model is used for an electron response. Furthermore, in order to facilitate simulations of highly-anisotropic microturbulence in the presence of strong magnetic shear and a magnetic X-point, a numerical algorithm utilizing a locally field-aligned multiblock coordinate system has been developed and implemented in the 5D finite-volume code COGENT. In this approach, the toroidal direction is divided into blocks, such that within each block, the cells are field-aligned and a non-matching grid interface is allowed at block boundaries. The toroidal angle corresponds to the “coarse” field-aligned coordinate, whereas the poloidal cross section, comprised of the radial and poloidal directions, is finely gridded to resolve short-scale perpendicular turbulence structures and to support accurate re-mapping (interpolation) at block boundaries. The 5D simulations explore cross-separatrix ion scale turbulence in the presence of a self-consistent radial electric field and address the effects of magnetic-shear stabilization in the X-point region.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1997

An Application of the Difference Potentials Method to Solving External Problems in CFD

Numerical solution of infinite-domain boundary-value problems requires some special techniques that would make the problem available for treatment on the computer. Indeed, the problem must be discretized in a way that the computer operates with only finite amount of information. Therefore, the original infinite-domain formulation must be altered and/or augmented so that on one hand the solution is not changed (or changed slightly) and on the other hand the finite discrete formulation becomes available. One widely used approach to constructing such discretizations consists of truncating the unbounded original domain and then setting the artificial boundary conditions (ABC's) at the newly formed external boundary. The role of the ABC's is to close the truncated problem and at the same time to ensure that the solution found inside the finite computational domain would be maximally close to (in the ideal case, exactly the same as) the corresponding fragment of the original infinite-domain solution. Let us emphasize that the proper treatment of artificial boundaries may have a profound impact on the overall quality and performance of numerical algorithms. The latter statement is corroborated by the numerous computational experiments and especially concerns the area of CFD, in which external problems present a wide class of practically important formulations. In this paper, we review some work that has been done over the recent years on constructing highly accurate nonlocal ABC's for calculation of compressible external flows. The approach is based on implementation of the generalized potentials and pseudodifferential boundary projection operators analogous to those proposed first by Calderon. The difference potentials method (DPM) by Ryaben'kii is used for the effective computation of the generalized potentials and projections. The resulting ABC's clearly outperform the existing methods from the standpoints of accuracy and robustness, in many cases noticeably speed up the multigrid convergence, and at the same time are quite comparable to other methods from the standpoints of geometric universality and simplicity of implementation.

Ryaben 'Kii, Victor S.↗
NASA NTRS1979

Numerical methods for problems involving the Drazin inverse

The objective was to try to develop a useful numerical algorithm for the Drazin inverse and to analyze the numerical aspects of the applications of the Drazin inverse relating to the study of homogeneous Markov chains and systems of linear differential equations with singular coefficient matrices. It is felt that all objectives were accomplished with a measurable degree of success.

Meyer, C. D., Jr.↗
NASA NTRS1993

High order hybrid numerical simulations of two dimensional detonation waves

In order to study multi-dimensional unstable detonation waves, a high order numerical scheme suitable for calculating the detailed transverse wave structures of multidimensional detonation waves was developed. The numerical algorithm uses a multi-domain approach so different numerical techniques can be applied for different components of detonation waves. The detonation waves are assumed to undergo an irreversible, unimolecular reaction A yields B. Several cases of unstable two dimensional detonation waves are simulated and detailed transverse wave interactions are documented. The numerical results show the importance of resolving the detonation front without excessive numerical viscosity in order to obtain the correct cellular patterns.

Cai, Wei↗
NASA NTRS2014

Towards Test Driven Development for Computational Science with pFUnit

Developers working in Computational Science & Engineering (CSE)/High Performance Computing (HPC) must contend with constant change due to advances in computing technology and science. Test Driven Development (TDD) is a methodology that mitigates software development risks due to change at the cost of adding comprehensive and continuous testing to the development process. Testing frameworks tailored for CSE/HPC, like pFUnit, can lower the barriers to such testing, yet CSE software faces unique constraints foreign to the broader software engineering community. Effective testing of numerical software requires a comprehensive suite of oracles, i.e., use cases with known answers, as well as robust estimates for the unavoidable numerical errors associated with implementation with finite-precision arithmetic. At first glance these concerns often seem exceedingly challenging or even insurmountable for real-world scientific applications. However, we argue that this common perception is incorrect and driven by (1) a conflation between model validation and software verification and (2) the general tendency in the scientific community to develop relatively coarse-grained, large procedures that compound numerous algorithmic steps.We believe TDD can be applied routinely to numerical software if developers pursue fine-grained implementations that permit testing, neatly side-stepping concerns about needing nontrivial oracles as well as the accumulation of errors. We present an example of a successful, complex legacy CSE/HPC code whose development process shares some aspects with TDD, which we contrast with current and potential capabilities. A mix of our proposed methodology and framework support should enable everyday use of TDD by CSE-expert developers.

pFUnit↗