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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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At least 181 records · Page 10
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DOE OSTI2023

ARKODE: A Flexible IVP Solver Infrastructure for One-step Methods

We describe the ARKODE library of one-step time integration methods for ordinary differential equation (ODE) initial-value problems (IVPs). In addition to providing standard explicit and diagonally implicit Runge–Kutta methods, ARKODE supports one-step methods designed to treat additive splittings of the IVP, including implicit-explicit (ImEx) additive Runge–Kutta methods and multirate infinitesimal (MRI) methods. We present the role of ARKODE within the SUNDIALS suite of time integration and nonlinear solver libraries, the core ARKODE infrastructure for utilities common to large classes of one-step methods, as well as its use of “time stepper” modules enabling easy incorporation of novel algorithms into the library. Numerical results show example problems of increasing complexity, highlighting the algorithmic flexibility afforded through this infrastructure, and include a larger multiphysics application leveraging multiple algorithmic features from ARKODE and SUNDIALS.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Extreme Scale Unstructured Adaptive CFD: From Multiphase Flow to Aerodynamic Flow Control

Understanding the flow of fluid, either liquid or gas, through and around solid bodies has challenged man since the dawn of scientific inquiry. Many of the great minds of science and math have progressively built up a hierarchy of fluid models. This report is concerned with the computational modeling of turbulent flow around aerodynamic bodies such as planes and wind turbines. In this case, viscous effects near the solid bodies create very thin boundary layers that yield highly anisotropic (gradients normal to the surface may be 10^6 times larger than gradients along the surface) solutions to the governing non-linear partial differential equations (PDE); the Navier-Stokes equations. Furthermore, turbulent flows develop extremely broad ranges of length and time scales.This disparity motivates the use of discretization methods capable of employing adaptivity and implicit time integration. The combination of these features (non-linear, anisotropy, adaptivity, andimplicit) dramatically raise the complexity of the discretization, posing large challenges to efficient scalable parallel implementation. However, through careful design, the more complex algorithms can provide great reductions in computational cost relative to simpler methods (e.g., Cartesian grids with explicit time integration) that are easier to mate efficiently to hardware. In this report, we not only describe our approach but we also address the fact that while complex algorithms may never be as efficient flop-for-flop as simple methods, in the important measure of science-per-core-hour, they can still win big by making complex features like adaptivity and implicit methods as efficient and scalable as possible.

42 ENGINEERING↗
DOE OSTI2019

Efficient IMEX Runge-Kutta Methods for Nonhydrostatic Dynamics

We develop new implicit-explicit Runge-Kutta (IMEX RK) methods for integrating horizontally explicit, vertically implicit (HEVI) partitionings of nonhydrostatic atmosphere models (HEVI models). These new methods, termed IMKG methods, are IMEX RK methods whose explicit part has optimal or near-optimal stability on the imaginary axis and whose implicit part is I-stable. A specialized stability region is presented for characterizing the stability of IMKG and other IMEX methods integrating HEVI models. Subsequently, we formulate two families of IMEX RK methods to enable deriving IMKG methods with a high explicit stage count for integrating HEVI models with large, stable time-steps. Here, we then derive a HEVI partitioning of the HOMME-NH nonhydrostatic model and use this model to compare the accuracy and efficiency of several IMKG methods with other IMEX RK methods from the literature.

58 GEOSCIENCES↗
DOE OSTI2023

A Conservative Finite-Volume Based Interface-Tracking Algorithm Using the Signed Distance Function

Methods for tracking an interface between two fluid phases are developed to ensure desired fluid properties, conservation, and stability are preserved in a finitevolume (FV) discretization. Here, the interface is tracked using a level set method where the signed distance function implicitly defines the interface. Marching methods are used to evaluate the value of the signed distance function, including a novel initialization method to initialize any implicit function to the signed distance function around sharp corners in the level set. Global conservation and consistency with a set of governing equations is enforced by a compression coefficient that measures the volumetric compression or expansion due to inaccuracies in the level set evaluation. A redistribution method is integrated into the volume correction to eliminate the small-cell instability while maintaining global conservation. This suite of methods is implemented and tested using static uniform velocity, and potential flow cases with multiple interface geometries. Results show these methods achieve up to second order accuracy, and are conservative. The application for these methods is intended to track the interface of a 3D printing filament in a finite-volume discretization of the all-speed Navier-Stokes equations.

42 ENGINEERING↗
DOE OSTI2016

Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG

Here, we discuss that the computational solution of the governing balance equations for mass, momentum, heat transfer and magnetic induction for resistive magnetohydrodynamics (MHD) systems can be extremely challenging. These difficulties arise from both the strong nonlinear, nonsymmetric coupling of fluid and electromagnetic phenomena, as well as the significant range of time- and length-scales that the interactions of these physical mechanisms produce. This paper explores the development of a scalable, fully-implicit stabilized unstructured finite element (FE) capability for 3D incompressible resistive MHD. The discussion considers the development of a stabilized FE formulation in context of the variational multiscale (VMS) method, and describes the scalable implicit time integration and direct-to-steady-state solution capability. The nonlinear solver strategy employs Newton–Krylov methods, which are preconditioned using fully-coupled algebraic multilevel preconditioners. These preconditioners are shown to enable a robust, scalable and efficient solution approach for the large-scale sparse linear systems generated by the Newton linearization. Verification results demonstrate the expected order-of-accuracy for the stabilized FE discretization. The approach is tested on a variety of prototype problems, that include MHD duct flows, an unstable hydromagnetic Kelvin–Helmholtz shear layer, and a 3D island coalescence problem used to model magnetic reconnection. Initial results that explore the scaling of the solution methods are also presented on up to 128K processors for problems with up to 1.8B unknowns on a CrayXK7.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

A blended continuous–discontinuous finite element method for solving the multi-fluid plasma model

The multi-fluid plasma model represents electrons, multiple ion species, and multiple neutral species as separate fluids that interact through short-range collisions and long-range electromagnetic fields. The model spans a large range of temporal and spatial scales, which renders the model stiff and presents numerical challenges. To address the large range of timescales, a blended continuous and discontinuous Galerkin method is proposed, where the massive ion and neutral species are modeled using an explicit discontinuous Galerkin method while the electrons and electromagnetic fields are modeled using an implicit continuous Galerkin method. This approach is able to capture large-gradient ion and neutral physics like shock formation, while resolving high-frequency electron dynamics in a computationally efficient manner. The details of the Blended Finite Element Method (BFEM) are presented. The numerical method is benchmarked for accuracy and tested using two-fluid one-dimensional soliton problem and electromagnetic shock problem. The results are compared to conventional finite volume and finite element methods, and demonstrate that the BFEM is particularly effective in resolving physics in stiff problems involving realistic physical parameters, including realistic electron mass and speed of light. The benefit is illustrated by computing a three-fluid plasma application that demonstrates species separation in multi-component plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2018

Solution of the Generalized Linear Boltzmann Equation for Multidimensional Stochastic Media

The generalized linear Boltzmann equation is a recently developed framework based on non-classical transport theory for modeling the expected value of particle flux in an arbitrary stochastic medium. Provided with a non-classical cross-section for a given statistical description of a medium, any transport problem in that medium may be solved. Previous work has only considered one-dimensional media without finite boundary conditions and discrete binary mixtures of materials. In this work the solution approach for the GLBE in multidimensional media with finite boundaries is outlined. The discrete ordinates method with an implicit discretization of the pathlength variable is used to leverage sweeping methods for the transport operator. In addition, several convenient approximations for non-classical cross-sections are introduced. The solution approach is verified against random realizations of a Gaussian process medium in a square enclosure.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2017

The Parareal Method for Solving Space-Dependent Neutron Kinetics Equations in Hexagonal-z Geometry

Neutron kinetics plays an important role in reactor safety and analysis. The backward Euler method is the most widely used time integration method in the calculation of space-dependent nuclear reactor kinetics. Xia (2006) used this method to discretize the derivative of flux, and adopted the analytic integration to discretize the delayed neutron precursor concentrations. Diagonally Implicit Runge-Kutta (DIRK) method owns high accuracy and excellent stability, and it was applied in the rectangle core for calculating core the nuclear reactor kinetics by Wenbo Zhao (2013). Recently, DIRK was applied to the neutron kinetics for hexagonal-z geometry application (CAI, 2016). As solving the neutron kinetics equations is very time-consuming and the number of available cores continues to increase with parallel architectures evolving, parallel algorithms need to be designed in order to effectively utilize the available resources. However, it is difficult to parallel in time axis because the later moment is strongly dependent on the previous moment. Some methods have been proposed to parallel in time algorithm over the last decades. One of the famous methods to realize time parallelism is the Parareal algorithm (Lions, 2001), which offers a promising approach to increase the degree of concurrency. More than one decade from this method first proposed, this method has been studied and applied in the many fields of applied mathematics, physics, chemistry, parallel computing, etc. Fischer (2014 applied the Parareal method in the field of the Navier-Stokes Equations. Baudron (2014) combined the Parareal method with finite element method to solve the neutron kinetics equations, where the Parareal method was implemented based on the implicit Euler method or method. In this paper, the Parareal method is studied in the hexagonal-z geometry with the help of DIRK developed for neutron kinetics equations (CAI, 2016)

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2022

Transient MOC with frequency transform and DSA on unstructured mesh

We present an implementation of the transient method of characteristics (MOC) with isotropic time derivatives, accelerated by diffusion synthetic acceleration (DSA). The fully implicit frequency transform method is used to solve the transient problem with analytic precursor integration. The code works on meshes composed of almost any of the commonly used non-curvilinear finite element types, and can handle the deformation of geometry in time-dependent transport calculations. We present results of a continuous Fourier analysis for the transient multigroup DSA problem, and representative benchmarking results are presented for the C5G7-TD benchmark in 2D showing reasonable performance and agreement compared to other codes. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2017

Acceleration of the IMplicit–EXplicit nonhydrostatic unified model of the atmosphere on manycore processors

We present the acceleration of an IMplicit-EXplicit (IMEX) nonhydrostatic atmospheric model on manycore processors such as graphic processing units (GPUs) and Intel's Many Integrated Core (MIC) architecture. IMEX time integration methods sidestep the constraint imposed by the Courant-Friedrichs-Lewy condition on explicit methods through corrective implicit solves within each time step. In this work, we implement and evaluate the performance of IMEX on manycore processors relative to explicit methods. Using 3D-IMEX at Courant number C = 15, we obtained a speedup of about 4x relative to an explicit time stepping method run with the maximum allowable C = 1. Moreover, the unconditional stability of IMEX with respect to the fast waves means the speedup can increase significantly with the Courant number as long as the accuracy of the resulting solution is acceptable. We show a speedup of 100x at C = 150 using 1D-IMEX to demonstrate this point. Several improvements on the IMEX procedure were necessary in order to outperform our results with explicit methods: (a) reducing the number of degrees of freedom of the IMEX formulation by forming the Schur complement, (b) formulating a horizontally explicit vertically implicit 1D-IMEX scheme that has a lower workload and better scalability than 3D-IMEX, (c) using high-order polynomial preconditioners to reduce the condition number of the resulting system, and (d) using a direct solver for the 1D-IMEX method by performing and storing LU factorizations once to obtain a constant cost for any Courant number. Without all of these improvements, explicit time integration methods turned out to be difficult to beat. We discuss in detail the IMEX infrastructure required for formulating and implementing efficient methods on manycore processors. Several parametric studies are conducted to demonstrate the gain from each of the abovementioned improvements. Lastly, we validate our results with standard benchmark problems in numerical weather prediction and evaluate the performance and scalability of the IMEX method using up to 4192 GPUs and 16 Knights Landing processors.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Higher-order temporal integration for the incompressible Navier–Stokes equations in bounded domains

Here, this paper compares and contrasts higher-order, semi-implicit temporal integration strategies for the incompressible Navier–Stokes methods based on spectral deferred corrections applied to certain gauge or auxiliary variable formulations of the equations. Particular focus is placed on the imposition of boundary conditions in the semi-implicit formulation, the accurate treatment of the pressure term, and the smoothness of the numerical solution for different formulations. The main result presented here is the formulation and numerical validation of a single-step, semi-implicit method called spectral deferred pressure corrections, which can in theory obtain arbitrary formal order of accuracy in time. Lastly, numerical results demonstrate up to eighth-order accuracy, scenarios where optimal temporal accuracy is attained, and scenarios where order reduction is observed due to time-dependent boundary conditions.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Nonequilibrium statistical mechanics and optimal prediction of partially-observed complex systems

Abstract Only a subset of degrees of freedom are typically accessible or measurable in real-world systems. As a consequence, the proper setting for empirical modeling is that of partially-observed systems. Notably, data-driven models consistently outperform physics-based models for systems with few observable degrees of freedom; e.g. hydrological systems. Here, we provide an operator-theoretic explanation for this empirical success. To predict a partially-observed system’s future behavior with physics-based models, the missing degrees of freedom must be explicitly accounted for using data assimilation and model parametrization. Data-driven models, in contrast, employ delay-coordinate embeddings and their evolution under the Koopman operator to implicitly model the effects of the missing degrees of freedom. We describe in detail the statistical physics of partial observations underlying data-driven models using novel maximum entropy and maximum caliber measures. The resulting nonequilibrium Wiener projections applied to the Mori–Zwanzig formalism reveal how data-driven models may converge to the true dynamics of the observable degrees of freedom. Additionally, this framework shows how data-driven models infer the effects of unobserved degrees of freedom implicitly, in much the same way that physics models infer the effects explicitly. This provides a unified implicit-explicit modeling framework for predicting partially-observed systems, with hybrid physics-informed machine learning methods combining both implicit and explicit aspects.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Strong Stability Preserving Integrating Factor Two-Step Runge-Kutta Methods

In this report, problems with components that feature significantly different time scales, where the stiff time-step restriction comes from a linear component, implicit-explicit (IMEX) methods alleviate this restriction if the concern is linear stability. However, when nonlinear non-inner-product stability properties are of interest, such as in the evolution of hyperbolic partial differential equations with shocks or sharp gradients, linear inner-product stability is no longer sufficient for convergence, and so strong stability preserving (SSP) methods are often needed. Where the SSP property is needed, IMEX SSP Runge-Kutta (SSP-IMEX) methods have very restrictive time-steps. An alternative to SSP-IMEX schemes is to adopt an integrating factor approach to handle the linear component exactly and step the transformed problem forward using some time-evolution method. The strong stability properties of integrating factor Runge-Kutta methods were established in Isherwood et al. (SIAM J Numer Anal 56(6):3276–3307, 2018), where it was shown that it is possible to define explicit integrating factor Runge-Kutta methods that preserve strong stability properties satisfied by each of the two components when coupled with forward Euler time-stepping. It was proved that the solution will be SSP if the transformed problem is stepped forward with an explicit SSP Runge-Kutta method that has non-decreasing abscissas. However, explicit SSP Runge-Kutta methods have an order barrier of p=4, and sometimes higher order is desired. In this work we consider explicit SSP two-step Runge-Kutta integrating factor methods to raise the order. We show that strong stability is ensured if the two-step Runge-Kutta method used to evolve the transformed problem is SSP and has non-decreasing abscissas. We find such methods up to eighth order and present their SSP coefficients. Adding a step allows us to break the fourth order barrier on explicit SSP Runge-Kutta methods; furthermore, our explicit SSP two-step Runge-Kutta methods with non-decreasing abscissas typically have larger SSP coefficients than the corresponding one-step methods. A selection of our methods are tested for convergence and demonstrate the design order. We also show, for selected methods, that the SSP time-step predicted by the theory is a lower bound of the allowable time-step for linear and nonlinear problems that satisfy the total variation diminishing (TVD) condition. We compare some of the non-decreasing abscissa SSP two-step Runge-Kutta methods to previously found methods that do not satisfy this criterion on linear and nonlinear TVD test cases to show that this non-decreasing abscissa condition is indeed necessary in practice as well as theory. We also compare these results to our SSP integrating factor Runge-Kutta methods designed in Isherwood et al. (2018).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

IMEX and exact sequence discretization of the multi-fluid plasma model

Multi-fluid plasma models, where an electron fluid is modeled in addition to multiple ion and neutral species as well as the full set of Maxwell's equations, are useful for representing physics beyond the scope of classic MHD. This advantage presents challenges in appropriately dealing with electron dynamics and electromagnetic behavior characterized by the plasma and cyclotron frequencies and the speed of light. For physical systems, such as those near the MHD asymptotic regime, this requirement drastically increases runtimes for explicit time integration even though resolving fast dynamics may not be critical for accuracy. Implicit time integration methods, with efficient solvers, can help to step over fast time-scales that constrain stability, but do not strongly influence accuracy. As an extension, Implicit-explicit (IMEX) schemes provide an additional mechanism to choose which dynamics are evolved using an expensive implicit solve or resolved using a fast explicit solve. In this study, in addition to IMEX methods we also consider a physics compatible exact sequence spatial discretization. Here, this combines nodal bases (H-Grad) for fluid dynamics with a set of vector bases (H-Curl and H-Div) for Maxwell's equations. This discretization allows for multi-fluid plasma modeling without violating Gauss' laws for the electric and magnetic fields. This initial study presents a discussion of the major elements of this formulation and focuses on demonstrating accuracy in the linear wave regime and in the MHD limit for both a visco-resistive and a dispersive ideal MHD problem.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2016

A comparative study of Rosenbrock-type and implicit Runge-Kutta time integration for discontinuous Galerkin method for unsteady 3D compressible Navier-Stokes equations

A comparative study of two classes of third-order implicit time integration schemes is presented for a third-order hierarchical WENO reconstructed discontinuous Galerkin (rDG) method to solve the 3D unsteady compressible Navier-Stokes equations: — 1) the explicit first stage, single diagonally implicit Runge-Kutta (ESDIRK3) scheme, and 2) the Rosenbrock-Wanner (ROW) schemes based on the differential algebraic equations (DAEs) of Index-2. Compared with the ESDIRK3 scheme, a remarkable feature of the ROW schemes is that, they only require one approximate Jacobian matrix calculation every time step, thus considerably reducing the overall computational cost. A variety of test cases, ranging from inviscid flows to DNS of turbulent flows, are presented to assess the performance of these schemes. Here, numerical experiments demonstrate that the third-order ROW scheme for the DAEs of index-2 can not only achieve the designed formal order of temporal convergence accuracy in a benchmark test, but also require significantly less computing time than its ESDIRK3 counterpart to converge to the same level of discretization errors in all of the flow simulations in this study, indicating that the ROW methods provide an attractive alternative for the higher-order time-accurate integration of the unsteady compressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗