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DOE OSTI2020

Partitioned exponential methods for coupled multiphysics systems

Multiphysics problems involving two or more coupled physical phenomena are ubiquitous in science and engineering. This work develops a new partitioned exponential approach for the time integration of multiphysics problems. After a possible semi-discretization in space, the class of problems under consideration is modeled by a system of ordinary differential equations where the right-hand side is a summation of two component functions, each corresponding to a given set of physical processes. The partitioned-exponential methods proposed herein evolve each component of the system via an exponential integrator, and information between partitions is exchanged via coupling terms. Here, the traditional approach to constructing exponential methods, based on the variation-of-constants formula, is not directly applicable to partitioned systems. Rather, our approach to developing new partitioned-exponential families is based on a general-structure additive formulation of the schemes. Two method formulations are considered, one based on a linear-nonlinear splitting of the right hand component functions, and another based on approximate Jacobians. The paper develops classical (non-stiff) order conditions theory for partitioned exponential schemes based on particular families of T-trees and B-series theory. Several practical methods of third order are constructed that extend the Rosenbrock-type and EPIRK families of exponential integrators. Several implementation optimizations specific to the application of these methods to reaction-diffusion systems are also discussed. Numerical experiments reveal that the new partitioned-exponential methods can perform better than traditional unpartitioned exponential methods on some problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Multi-element flow-driven spectral chaos (ME-FSC) method for uncertainty quantification of dynamical systems

The flow-driven spectral chaos (FSC) is a recently developed method for tracking and quantifying uncertainties in the long-time response of stochastic dynamical systems using the spectral approach. The method uses a novel concept called enriched stochastic flow maps as a means to construct an evolving finite-dimensional random function space that is both accurate and computationally efficient in time. In this paper, we present a multi-element version of the FSC method (the ME-FSC method for short) to tackle (mainly) those dynamical systems that are inherently discontinuous over the probability space. In ME-FSC, the random domain is partitioned into several elements, and then the problem is solved separately on each random element using the FSC method. Subsequently, results are aggregated to compute the probability moments of interest using the law of total probability. To demonstrate the effectiveness of the ME-FSC method in dealing with discontinuities and long-time integration of stochastic dynamical systems, four representative numerical examples are presented in this paper, including the Van-der-Pol oscillator problem and the Kraichnan-Orszag three-mode problem. Results show that the ME-FSC method is capable of solving problems that have strong nonlinear dependencies over the probability space, both reliably and at low computational cost.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Implicit–explicit multirate infinitesimal stage-restart methods

Implicit–Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is split into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems split into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of “fast” IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit–explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further split in an IMEX fashion. This new class, which we call implicit–explicit multirate infinitesimal stage-restart (IMEX-MRI-SR), both improves upon the previous implicit–explicit multirate infinitesimal generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods by allowing for far easier creation of new embedded methods, and extends multirate exponential Runge Kutta (MERK) methods by allowing the fast-changing dynamics to be nonlinear and the methods to be implicit. We leverage GARK theory to derive conditions for orders of accuracy up to four, and we provide second- and third-order accurate example methods, which are the first known embedded MRI methods with IMEX structure. We then perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step settings.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗

DOE OSTI2021

Studies and optimization of hybrid X-pinches

Studies were carried out on the hybrid X-pinch (HXP) configuration for several wire materials using a 40ns risetime, 300-kA pulsed-power generator with the goal of optimizing the HXP X-ray burst parameters for different applications. The gap distance between the conical electrodes of the hybrid X-pinch was varied from 0.5 to 5 mm for Al, Ag, Ti, and Mo using different wire diameters to maintain a consistent mass per unit length from one material to the next. It was found that 0.5 - 1 mm gap spacing was optimal to obtain a single X-ray burst, while the number of X-ray bursts increased with gap spacing for all materials at a rate of 1-2/mm. Time consistency studies of the first X-ray burst were carried out for Ti wire by changing the gap distance and wire diameter while observing the time of occurrence of the X-ray bursts. Here, it was found that a 40μm Ti wire load with a 3mm electrode gap spacing had the highest probability to have the first X-ray burst reproducibly within a 2 ns time window. Time-resolved and time-integrated diagnostics provided information about the source size, time of occurrence, intensity, and the number of the X-ray bursts.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1971

Double phase-lock loop with rapid transient response - A concept

Integration loop, with narrow bandwidth and long response time, integrates phase estimate of sampling loop, resulting in slower but less jittery response to phase step changes. Sampling loop has wide bandwidth and quick response to phase steps of burst sync interval.

Balcewicz, J. F.↗

NASA NTRS1971

Digital aspect clock

Digital clock precisely set and reset by pulses from a solar sensor, combined with a logic system, provides accurate time-sector division of spin-stabilized satellite. Integral times for viewing physical phenomena from various directions are equal and mean angles of viewing can be determined.

Bartley, W. C.↗

NASA NTRS1993

A small-scale turbulence model

A previously derived analytical model for the small-scale structure of turbulence is reformulated in such a way that the energy spectrum may be computed. The model is an ensemble of two-dimensional (2D) vortices with internal spiral structure, each stretched by an axially symmetric strain flow. Stretching and differential rotation produce an energy cascade to smaller scales in which the stretching represents the effect of instabilities and the spiral structure is the source of dissipation at the end of the cascade. The energy spectrum of the resulting flow may be expressed as a time integration involving only the enstrophy spectrum of the time evolving 2D cross section flow, which may be obtained numerically. Examples are given in which a k exp -5/3 spectrum is obtained by this method. The k exp -5/3 inertial range spectrum is shown to be related to the existence of a self-similar enstrophy preserving range in the 2D enstrophy spectrum. The results are found to be insensitive to time dependence of the strain rate, including even intermittent on-or-off strains.

Lundgren, T. S.↗

NASA NTRS1992

A small-scale turbulence model

A model for the small-scale structure of turbulence is reformulated in such a way that it may be conveniently computed. The model is an ensemble of randomly oriented structured two dimensional vortices stretched by an axially symmetric strain flow. The energy spectrum of the resulting flow may be expressed as a time integral involving only the enstrophy spectrum of the time evolving two-dimensional cross section flow, which may be obtained numerically. Examples are given in which a k(exp -5/3) spectrum is obtained by this method without using large wave number asymptotic analysis. The k(exp -5/3) inertial range spectrum is shown to be related to the existence of a self-similar enstrophy preserving range in the two-dimensional enstrophy spectrum. The results are insensitive to time dependence of the strain-rate, including even intermittent on-or-off strains.

Lundgren, T. S.↗

NASA NTRS2009

Local-in-Time Adjoint-Based Method for Optimal Control/Design Optimization of Unsteady Compressible Flows

.We study local-in-time adjoint-based methods for minimization of ow matching functionals subject to the 2-D unsteady compressible Euler equations. The key idea of the local-in-time method is to construct a very accurate approximation of the global-in-time adjoint equations and the corresponding sensitivity derivative by using only local information available on each time subinterval. In contrast to conventional time-dependent adjoint-based optimization methods which require backward-in-time integration of the adjoint equations over the entire time interval, the local-in-time method solves local adjoint equations sequentially over each time subinterval. Since each subinterval contains relatively few time steps, the storage cost of the local-in-time method is much lower than that of the global adjoint formulation, thus making the time-dependent optimization feasible for practical applications. The paper presents a detailed comparison of the local- and global-in-time adjoint-based methods for minimization of a tracking functional governed by the Euler equations describing the ow around a circular bump. Our numerical results show that the local-in-time method converges to the same optimal solution obtained with the global counterpart, while drastically reducing the memory cost as compared to the global-in-time adjoint formulation.

Yamaleev, N. K.↗

NASA NTRS2013

Visible-Infrared Hyperspectral Image Projector

The VisIR HIP generates spatially-spectrally complex scenes. The generated scenes simulate real-world targets viewed by various remote sensing instruments. The VisIR HIP consists of two subsystems: a spectral engine and a spatial engine. The spectral engine generates spectrally complex uniform illumination that spans the wavelength range between 380 nm and 1,600 nm. The spatial engine generates two-dimensional gray-scale scenes. When combined, the two engines are capable of producing two-dimensional scenes with a unique spectrum at each pixel. The VisIR HIP can be used to calibrate any spectrally sensitive remote-sensing instrument. Tests were conducted on the Wide-field Imaging Interferometer Testbed at NASA s Goddard Space Flight Center. The device is a variation of the calibrated hyperspectral image projector developed by the National Institute of Standards and Technology in Gaithersburg, MD. It uses Gooch & Housego Visible and Infrared OL490 Agile Light Sources to generate arbitrary spectra. The two light sources are coupled to a digital light processing (DLP(TradeMark)) digital mirror device (DMD) that serves as the spatial engine. Scenes are displayed on the DMD synchronously with desired spectrum. Scene/spectrum combinations are displayed in rapid succession, over time intervals that are short compared to the integration time of the system under test.

Bolcar, Matthew↗

NASA NTRS2014

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used.

Nielsen, Tanner B.↗

DOE OSTI2024

Asymptotic-preserving gyrokinetic implicit particle-orbit integrator for arbitrary electromagnetic fields

We extend the asymptotic preserving and energy conserving time integrator for charged-particle motion developed in Ricketson and Chacón (2020) to include finite Larmor-radius (FLR) effects in the presence of electric-field length-scales comparable to the particle gyro-radius (the gyro-kinetic limit). We introduce two modifications to the earlier scheme. The first is the explicit gyro-averaging of the electric field at the half time-step, along with an analogous modification to the current deposition, which we show preserves total energy conservation in implicit PIC schemes. The number of gyrophase samples is chosen adaptively, ensuring proper averaging for large timesteps and the recovery of full-orbit dynamics in the small time-step limit. The second modification is an alternating large and small time-step strategy that ensures the particle trajectory samples gyrophases evenly. We show that this strategy relaxes the time-step restrictions on the scheme, allowing even larger speed-ups than previously achievable. We demonstrate the new method with several single-particle motion tests in a variety of electromagnetic field configurations featuring gyro-scale variation in the electric field. Finally, the results demonstrate the advertised ability to capture FLR effects accurately even when significantly stepping over the gyration time-scale.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1998

Solar System Studies with the Space Infrared Telescope Facility (SIRTF)

SIRTF (Space Infrared Telescope Facility) is the final element in NASA's 'Great Observatories' program. It consists of an 85-cm cryogenically-cooled observatory for infrared astronomy from space. SIRTF is scheduled for launch in late 2001 or early 2002 on a Delta rocket into a heliocentric orbit trailing the Earth. Data from SIRTF will be processed and disseminated to the community through the SIRTF Science Center (SSC) located at the Infrared Processing and Analysis Center (IPAC) at Caltech. Some 80/% of the total observing time (estimated at a minimum of 7500 hours of integration time per year for the mission lifetime of about 4 years) will be available to the scientific community at large through a system of refereed proposals. Three basic instruments are located in the SIRTF focal plane. The Multiband Imaging Photometer (MIPS), the Infrared Array Camera (IRAC), and the Infrared Spectrometer (IRS), taken together, provide imaging and spectroscopy from 3.5 to 160 microns. Among the solar system studies suited to SIRTF are the following: 1) spectroscopy and radiometry of small bodies from the asteroid main belt, through the Trojan clouds, to the Kuiper Disk; 2) dust distribution in the zodiacal cloud and the Earth's heliocentric dust ring; 3) spectroscopy and radiometry of comets; and 4) spectroscopy and radiometry of planets and their satellites. Searches for, and studies of dust disks around other stars, brown dwarfs, and superplanets will also be conducted with SIRTF. The SORTIE web site (http://ssc.ipac.caltech.edu/sirtf) contains important details and documentation on the project, the spacecraft, the telescope, instruments, and observing procedures. A community-wide workshop for solar system studies with SIRTF is in the planning stages by the author and Martha S. Hanner for the summer of 1999.

Cruikshank, Dale P.↗

DOE OSTI2022

Fast Solution of Fully Implicit Runge--Kutta and Discontinuous Galerkin in Time for Numerical PDEs, Part II: Nonlinearities and DAEs

Fully implicit Runge--Kutta (IRK) methods have many desirable accuracy and stability properties as time integration schemes, but high-order IRK methods are not commonly used in practice with large-scale numerical PDEs because of the difficulty of solving the stage equations. This paper introduces a theoretical and algorithmic framework for solving the nonlinear equations that arise from IRK methods (and discontinuous Galerkin discretizations in time) applied to nonlinear numerical PDEs, including PDEs with algebraic constraints. Several new linearizations of the nonlinear IRK equations are developed, offering faster and more robust convergence than the often-considered simplified Newton, as well as an effective preconditioner for the true Jacobian if exact Newton iterations are desired. Inverting these linearizations requires solving a set of block 2 x 2 systems. Under quite general assumptions, it is proven that the preconditioned 2 x 2 operator's condition number is bounded by a small constant close to one, independent of the spatial discretization, spatial mesh, and time step, and with only weak dependence on the number of stages or integration accuracy. Moreover, the new method is built using the same preconditioners needed for backward Euler-type time stepping schemes, so can be readily added to existing codes. The new methods are applied to several challenging fluid flow problems, including the compressible Euler and Navier--Stokes equations, and the vorticity-streamfunction formulation of the incompressible Euler and Navier--Stokes equations. Up to 10th-order accuracy is demonstrated using Gauss IRK, while in all cases fourth-order Gauss IRK requires roughly half the number of preconditioner applications as required by standard Singly diagonally implicit Runge--Kutta methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Asymptotic-preserving dynamical low-rank method for the stiff nonlinear Boltzmann equation

In kinetic theory, numerically solving the full Boltzmann equation is extremely expensive. This is because the Boltzmann collision operator involves a high-dimensional, nonlinear integral that must be evaluated at each spatial grid point and every time step. The challenge becomes even more pronounced in the fluid (strong collisionality) regime, where the collision operator exhibits strong stiffness, causing explicit time integrators to impose severe stability restrictions. In this paper, we propose addressing this problem through a dynamical low-rank (DLR) approximation. The resulting algorithm requires evaluating the Boltzmann collision operator only r 2 times, where r, the rank of the approximation, is much smaller than the number of spatial grid points. We propose a novel DLR integrator, called the XL integrator, which reduces the number of steps compared to the available alternatives (such as the projector splitting or basis update & Galerkin (BUG) integrator). For a class of problems including the Boltzmann collision operator which enjoys a separation property between physical and velocity space, we further propose a specialized version of the XL integrator, called the sXL integrator. This version requires solving only one differential equation to update the low-rank factors. Furthermore, the proposed low-rank schemes are asymptotic-preserving, meaning they can capture the asymptotic fluid limit in the case of strong collisionality. Our numerical experiments demonstrate the efficiency and accuracy of the proposed methods across a wide range of regimes, from non-stiff (kinetic) to stiff (fluid).

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

A linearity preserving nodal variation limiting algorithm for continuous Galerkin discretization of ideal MHD equations

A stabilized continuous Galerkin (CG) method for magnetohydrodynamics (MHD) is presented herein. Ideal, compressible inviscid MHD equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi-discrete scheme. Stabilization is then introduced to the semi-discrete method in a strategy that follows the algebraic flux correction paradigm. This involves adding some artificial diffusion to the high order, semi-discrete method and mass lumping in the time derivative term. The result is a low order method that provides local extremum diminishing properties for hyperbolic systems. The difference between the low order method and the high order method is scaled element-wise using a limiter and added to the low order scheme. The limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy. The stabilization also involves an optional consistent background high order dissipation that reduces phase errors. The resulting stabilized scheme is a semi-discrete method that can be applied to inviscid shock MHD problems and may be even extended to resistive and viscous MHD problems. To satisfy the divergence free constraint of the MHD equations, we add parabolic divergence cleaning to the system. Various time integration methods can be used to discretize the scheme in time. We demonstrate the robustness of the scheme by solving several shock MHD problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Verification of a fully implicit particle-in-cell method for the  v ∥ -formalism of electromagnetic gyrokinetics in the XGC code

A fully implicit particle-in-cell method for handling the v ∥ -formalism of electromagnetic gyrokinetics has been implemented in XGC. By choosing the v ∥ -formalism, here we avoid introducing the nonphysical skin terms in Ampère's law, which are responsible for the well-known “cancellation problem” in the p ∥ -formalism. The v ∥ -formalism, however, is known to suffer from a numerical instability when explicit time integration schemes are used due to the appearance of a time derivative in the particle equations of motion from the inductive component of the electric field. Here, using the conventional δf scheme, we demonstrate that our implicitly discretized algorithm can provide numerically stable simulation results with accurate dispersive properties. We verify the algorithm using a test case for shear Alfvén wave propagation in addition to a case demonstrating the ion temperature gradient-kinetic ballooning mode (ITG-KBM) transition. The ITG-KBM transition case is compared to results obtained from other δf gyrokinetic codes/schemes, whose verification has already been archived in the literature.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Summary of Results for Idaho Accelerator Center (IAC) Experiments with Compact Electron Permanent Magnet Spectrometer (CEPMS)

The Compact Electron Permanent Magnet Spectrometer (CEPMS) shown in Figure 2 consists of collimation at the electron beam entrance, a vacuum chamber for beam transport and a removable 60 degree sector magnet constructed by SABR Enterprises (Ref 2) that is used to momentum analyze the incoming electron beam. There are electron detection planes located perpendicular to straight through and the 60 degree trajectory ports. The planes are designed to accept phosphor screens for time integrated measurements. The screens can be replaced by scintillator for time resolved measurements using gated or streak cameras. The length of the spectrometer along the beamline without the collimator is about 400mm (14”). The spectrometer requires no power supply with controls for cycling or cooling for the magnet. The entire assembly, with magnet installed, weighs about 35kg (75lbs) Figure 3 depicts central and extreme trajectories through the spectrometer for various electron energies. The chamber accommodates a +/-50mm wide detection field relative to the center trajectory. Trajectories with larger momentum (energy) are bent the least. Since the permanent magnet slides off and on with reference to the upstream edge of the vacuum chamber, it can be adjusted to various positions to slightly increase or decrease the central energy thereby changing the range of available energies. Figure 4 shows the effect of changing the location of the permanent magnet on the electron trajectory. A small inward change in r produces a longer trajectory in the magnet therefore, the same energy trajectory is bent more.

43 PARTICLE ACCELERATORS↗