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DOE OSTI2021

A DG-IMEX Method for Two-moment Neutrino Transport: Nonlinear Solvers for Neutrino–Matter Coupling

Neutrino-matter interactions play an important role in core-collapse supernova (CCSN) explosions as they contribute to both lepton number and/or four-momentum exchange between neutrinos and matter, and thus act as the agent for neutrino-driven explosions. Due to the multiscale nature of neutrino transport in CCSN simulations, an implicit treatment of neutrino-matter interactions is desired, which requires solutions of coupled nonlinear systems in each step of the time integration scheme. In this paper we design and compare nonlinear iterative solvers for implicit systems with energy coupling neutrino-matter interactions commonly used in CCSN simulations. Specifically, we consider electron neutrinos and antineutrinos, which interact with static matter configurations through the Bruenn 85 opacity set. The implicit systems arise from the discretization of a nonrelativistic two-moment model for neutrino transport, which employs the discontinuous Galerkin (DG) method for phase-space discretization and an implicit-explicit (IMEX) time integration scheme. In the context of this DG-IMEX scheme, we propose two approaches to formulate the nonlinear systems — a coupled approach and a nested approach. For each approach, the resulting systems are solved with Anderson-accelerated fixedpoint iteration and Newton’s method. The performance of these four iterative solvers has been compared on relaxation problems with various degree of collisionality, as well as proto-neutron star deleptonization problems with several matter profiles adopted from spherically symmetric CCSN simulations. Here, numerical results suggest that the nested Anderson-accelerated fixed-point solver is more efficient than other tested solvers for solving implicit nonlinear systems with energy coupling neutrino-matter interactions.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2025

Stage-local partitioned two-step runge-kutta methods for large systems of ordinary differential equations

We introduce stage-local partitioned two-step Runge-Kutta methods are an extension of standard two-step Runge-Kutta methods, which are an alternative to the standard additive two-step Runge-Kutta methods currently existing in the literature. Furthermore, these new schemes are designed with an eye towards truly N-partitioned systems and leverage local stage approximations to make several computationally interesting approximations viable. Specifically, the focus on local stage approximations makes possible the construction of truly asynchronous schemes, in the parallel sense, possible. In addition, we show that an implicit-explicit approach to these schemes can lead to methods that require the inversion of only local nonlinear systems.

Applied Dynamical Systems↗

DOE OSTI2022

The Schwarz alternating method for transient solid dynamics

Abstract In our earlier work, we formulated the Schwarz alternating method as a means for concurrent multiscale coupling in finite deformation solid mechanics for quasi‐static problems. Herein, we advance this method for the study of transient dynamic multiscale solid mechanics problems where information is exchanged back and forth between small and large scales. The extension to dynamics relies on the notion of a global time stepper. Within each global time step, the subdomains are coupled by the standard Schwarz iterative process. Remarkably, each subdomain can use its own time step or even its own time integrator to advance its solution in time, provided that they synchronize at each global time step. We study the performance of the Schwarz method on several examples designed for this purpose. Our numerical experiments demonstrate that the method is capable of coupling regions with different mesh resolutions, different element types, and different time integration schemes (e.g., implicit and explicit), all without introducing any artifacts that afflict other coupling methods for transient dynamics. Finally, we apply the dynamic Schwarz alternating method to the simulation of a bolted joint subjected to dynamic loading, as a demonstration of the performance of the method in a realistic scenario.

Mota, Alejandro↗

DOE OSTI2026

thornado+FLASH-X: A Hybrid Discontinuous Galerkin–Implicit-explicit and Finite-volume Framework for Neutrino-radiation Hydrodynamics in Core-collapse Supernovae

We present neutrino-transport algorithms implemented in the toolkit for high-order neutrino-radiation hydrodynamics (thornado) and their coupling to self-gravitating hydrodynamics within the adaptive mesh refinement–based multiphysics simulation framework FLASH-X. thornado, developed primarily for simulations of core-collapse supernovae (CCSNe), employs a spectral, six-species two-moment formulation with algebraic closure and special-relativistic observer corrections accurate to $\mathcal{O}(v/c)$, and uses discontinuous Galerkin (DG) methods for phase-space discretization combined with implicit-explicit time stepping. A key development is a nonlinear neutrino–matter coupling algorithm based on nested fixed-point iteration with Anderson acceleration, enabling fully implicit treatment of collisional processes, including energy-coupling interactions such as neutrino–electron scattering and pair production. Coupling to finite-volume (FV) hydrodynamics is achieved through a hybrid DG-FV representation of the fluid variables and operator-split evolution within FLASH-X. The implementation is verified using basic transport tests with idealized opacities and relaxation and deleptonization problems with tabulated microphysics. Spherically symmetric CCSN simulations demonstrate accuracy and robustness of the coupled scheme, including close agreement with the CCSN simulation code Chimera. An axisymmetric CCSN simulation further demonstrates the viability of DG-based neutrino transport for multidimensional supernova modeling within FLASH-X. thornado’s neutrino-transport solver is GPU-enabled using OpenMP offloading or OpenACC, and all CCSN applications included in this work use the GPU implementation. Together, these results establish a foundation for future enhancements in physics fidelity, numerical algorithms, and computational performance, for increasingly realistic large-scale CCSN simulations.

Endeve, Eirik [Oak Ridge National Laboratory (ORNL↗

DOE OSTI2022

On numerical energy conservation for an implicit particle-in-cell method coupled with a binary Monte-Carlo algorithm for Coulomb collisions

We report conventional particle-in-cell (PIC) methods suffer from enhanced numerical heating (explicit PIC) or cooling (semi-implicit PIC) when coupled with a binary Monte-Carlo algorithm for Coulomb collisions. In this work, a fully-implicit θ-PIC scheme (with adjustable time-biasing parameter 1/2 ≤ θ ≤) is considered. The discrete change in energy of a closed system after a time step for this scheme scales with (1/2 - θ)C θ , where C θ is a positive definite quantity that depends on the frequency spectrum of the energy in the fields. Collisions lead to additional energy in the field fluctuations associated with high-frequency light waves produced by a numerical Bremsstrahlung process, which can result in a large increase in the numerical cooling rate for θ > 1/2. However, for θ = 1/2, energy is exactly conserved. The energy in the field fluctuations on long time scales agrees with that calculated using the equipartition theorem for a classical system in thermodynamic equilibrium.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2022

Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency

Here, we consider a kinetic description of multi-species gas mixture modeled with Bhatnagar-Gross-Krook (BGK) collision operators, in which the collision frequency varies not only in time and space but also with the microscopic velocity. In this model, the Maxwellians typically used in standard BGK operators are replaced by a generalization of such target functions, which are defined by a variational procedure. In this paper we present a numerical method for simulating this model, which uses an Implicit-Explicit (IMEX) scheme to minimize a certain potential function, mimicking the Lagrange functional that appears in the theoretical derivation. We show that theoretical properties such as conservation of mass, total momentum and total energy as well as positivity of the distribution functions are preserved by the numerical method, and illustrate its usefulness and effectiveness with numerical examples.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

Unified Approach to Implicit and Explicit Solvent Simulations of Electrochemical Reaction Energetics

One of the major open challenges in ab initio simulations of the electrochemical interface is the determination of electrochemical barriers under a constant driving force. Existing methods to do so include extrapolation techniques based on fully explicit treatments of the electrolyte, as well as implicit solvent models which allow for a continuous variation in electrolyte charge. Emerging hybrid continuum models have the potential to revolutionize the field, since they account for the electrolyte with little computational cost while retaining some explicit electrolyte, representing a “best of both worlds” method. In this work, we present a unified approach to determine reaction energetics from fully explicit, implicit, and hybrid treatments of the electrolyte based on a new multicapacitor model of the electrochemical interface. A given electrode potential can be achieved by a variety of interfacial structures; a crucial insight from this work is that the effective surface charge gives a good proxy of the local potential, the true driving force of electrochemical processes. In contrast, we show that the traditionally considered work function gives rise to multivalued functions depending on the simulation cell size. Furthermore, we show that the reaction energetics are largely insensitive to the countercharge distribution chosen in hybrid implicit/explicit models, which means that any of the myriad implicit electrolyte models can be equivalently applied. Furthermore, this work thus paves the way for the accurate treatment of ab initio reaction energetics of general surface electrochemical processes using both implicit and explicit electrolytes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2017

Rythmos: Solution and Analysis Package for Differential-Algebraic and Ordinary-Differential Equations

Time integration is a central component for most transient simulations. It coordinates many of the major parts of a simulation together, e.g., a residual calculation with a transient solver, solution with the output, various operator-split physics, and forward and adjoint solutions for inversion. Even though there is this variety in these transient simulations, there is still a common set of algorithms and procedures to progress transient solutions for ordinary-differential equations (ODEs) and differential-alegbraic equations (DAEs). Rythmos is a collection of these algorithms that can be used for the solution of transient simulations. It provides common time-integration methods, such as Backward and Forward Euler, Explicit and Implicit Runge-Kutta, and Backward-Difference Formulas. It can also provide sensitivities, and adjoint components for transient simulations. Rythmos is a package within Trilinos, and requires some other packages (e.g., Teuchos and Thrya) to provide basic time-integration capabilities. It also can be coupled with several other Trilinos packages to provide additional capabilities (e.g., AztecOO and Belos for linear solutions, and NOX for non-linear solutions). The documentation is broken down into three parts: Theory Manual, User's Manual, and Developer's Guide. The Theory Manual contains the basic theory of the time integrators, the nomenclature and mathematical structure utilized within Rythmos, and verification results demonstrating that the designed order of accuracy is achieved. The User's Manual provides information on how to use the Rythmos, description of input parameters through Teuchos Parameter Lists, and description of convergence test examples. The Developer's Guide is a high-level discussion of the design and structure of Rythmos to provide information to developers for the continued development of capabilities. Details of individual components can be found in the Doxygen webpages.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

The Challenges of Safe Troubleshooting Work

Troubleshooting work presents electrical and other workers with a challenging combination of physical hazards, working conditions, and time pressure, which can lead to unwanted outcomes if not carefully managed. Summaries of several incidents in which workers were injured or at risk of injury while performing troubleshooting work are presented, identifying organizational weaknesses and error precursors that contributed to each incident. The primary challenges include: Deranged equipment. Equipment that needs troubleshooting is not in a normal operating condition. Actions that are safe when the equipment is in a normal state may not be safe in the deranged state. Work planning and control. The steps taken in troubleshooting are most often determined by the results of the immediately previous diagnostic test, making effective work planning challenging. Multiple types of hazards. Most equipment will present a troubleshooting worker with several types of hazards, including hazardous energy as defined in 29 CFR 1910.147. Portions of the troubleshooting activity may be infeasible without these hazards present. Time pressure. Restoring operation of failed equipment often involves an explicit or implicit sense of urgency. There are effective methods for addressing each challenge, most of which require a combination of advance preparation and management commitment.

Mertz, David E.↗

DOE OSTI2016

Fourier-spectral element approximation of the ion–electron Braginskii system with application to tokamak edge plasma in divertor configuration

Due to the extreme conditions required to produce energy by nuclear fusion in tokamaks, simulating the plasma behavior is an important but challenging task. We focus on the edge part of the plasma, where fluid approaches are probably the best suited, and our approach relies on the Braginskii ion–electron model. Assuming that the electric field is electrostatic, this yields a set of 10 strongly coupled and non-linear conservation equations that exhibit multiscale and anisotropy features. The computational domain is a torus of complex geometrical section, that corresponds to the divertor configuration, i.e. with an “X-point” in the magnetic surfaces. To capture the complex physics that is involved, high order methods are used: The time-discretization is based on a Strang splitting, that combines implicit and explicit high order Runge–Kutta schemes, and the space discretization makes use of the spectral element method in the poloidal plane together with Fourier expansions in the toroidal direction. The paper thoroughly describes the algorithms that have been developed, provides some numerical validations of the key algorithms and exhibits the results of preliminary numerical experiments. In particular, we point out that the highest frequency of the system is intermediate between the ion and electron cyclotron frequencies.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2014

A projection preconditioner for solving the implicit immersed boundary equations

This study presents a method for solving the linear semi-implicit immersed boundary equations which avoids the severe time step restriction presented by explicit-time methods. The Lagrangian variables are eliminated via a Schur complement to form a purely Eulerian saddle point system, which is preconditioned by a projection operator and then solved by a Krylov subspace method. From the viewpoint of projection methods, we derive an ideal preconditioner for the saddle point problem and compare the efficiency of a number of simpler preconditioners that approximate this perfect one. For low Reynolds number and high stiffness, one particular projection preconditioner yields an efficiency improvement of the explicit IB method by a factor around thirty. Substantial speed-ups over explicit-time method are achieved for Reynolds number below 100. In conclusion, this speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced.

42 ENGINEERING↗

DOE OSTI2023

Explicit dispersion relations for warm fluid waves in a uniform plasma (invited)

Classical dispersion relations for waves in a fluid plasma are expressed as implicit functions of wave frequency and wave- number. The implicit dispersion relation must in general be solved numerically. This work introduces an astute method for obtaining an explicit dispersion relation for waves in a fluid plasma. The explicit expression has an advantage of providing eigenmodes without numerical computations and giving a dispersion relation more detailed than the implicit expression. As in the case of cold waves, the wavenumber for an arbitrary frequency can be directly obtained from the explicit formula. The explicit formula also enables an accurate evaluation of finite-temperature effects on a dispersion relation even near the cyclotron resonance, where a numerical analysis of the implicit relation is impractical. As a result, the analytic formula can be used to investigate temperature effects on electromagnetic waves.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2016

Semi-implicit time integration of atmospheric flows with characteristic-based flux partitioning

Here, this paper presents a characteristic-based flux partitioning for the semi-implicit time integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time scale is significantly faster than the advective scale, yet it is typically not relevant to atmospheric and weather phenomena. The acoustic and advective components of the hyperbolic flux are separated in the characteristic space. High-order, conservative additive Runge-Kutta methods are applied to the partitioned equations so that the acoustic component is integrated in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale. Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared with that of explicit time integration.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2020

Reconciling Work Functions and Adsorption Enthalpies for Implicit Solvent Models: A Pt (111)/Water Interface Case Study

Implicit solvent models are a computationally efficient method of representing solid/liquid interfaces prevalent in electrocatalysis, energy storage, and materials science. However, electronic structure changes induced at the metallic surface by the dielectric continuum are not fully understood. To address this, we perform DFT calculations for the Pt(111)/water interface, in order to compare Poisson–Boltzmann continuum solvation methods with ab initio molecular dynamics (AIMD) simulations of explicit solvent. We show that the implicit solvent cavity can be parametrized in terms of the electric dipole moment change at the equilibrated explicit Pt/water interface to obtain the potential of zero charge (PZC). We also compare the accuracy of aqueous enthalpies of adsorption of phenol on Pt(111) using geometry and charge density based dielectric cavitation methods. The ability to parametrize the cavity according to individual atoms, as afforded in the geometry based approach, is key to obtaining accurate enthalpy changes of adsorption under aqueous conditions. Additionally, we show that the electronic structure changes induced by explicit solvent and our proposed implicit solvent parametrization scheme yield comparable density difference profiles and d-band projected density of states. We therefore demonstrate the capability of implicit solvent approaches to capture both the energetics of adsorption processes and the main electronic effects of aqueous solvent on the metallic surface. Therefore, this work provides a scheme for computationally efficient simulations of interfacial processes for applications in areas such as heterogeneous catalysis and electrochemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2023

Efficient Multigrid Reduction-in-Time for Method-of-Lines Discretizations of Linear Advection

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. In this report we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2013

Ab initio calculation of the deprotonation constants of an atomistically defined nanometer-sized, aluminium hydroxide oligomer

Aluminium possesses significant and diverse chemistry. Numerous compounds have been defined, and the elucidation of their chemistry is of significant geochemical interest. In this paper, a brucite-like, eight-aluminium aqueous cluster is modelled with density functional theory to identify its primary site of deprotonation and the associated pK(a) constant using both explicit (a full first solvent shell) and implicit solvent. Two methods for calculating the pK(a) are compared. We found that a bond density approach is better than a direct energy calculation for ions with large charge and high symmetry. The terminal aluminium atoms have equatorial ligated waters that in solvent have one long O-H bond. This site is more reactive than any of the other protons on the particle. Insights into the experimental crystal structure and Bader's Atoms in Molecules density analysis are presented as routes to reduce the computational time required for the identification of protonation sites.

AQUEOUS SOLVATION↗

DOE OSTI2022

Machine learning of consistent thermodynamic models using automatic differentiation

In this study, we propose a data-driven method to describe consistent equations of state (EOS) for arbitrary systems. Complex EOS are traditionally obtained by fitting suitable analytical expressions to thermophysical data. A key aspect of EOS is that the relationships between state variables are given by derivatives of the system free energy. In this work, we model the free energy with an artificial neural network and utilize automatic differentiation to directly learn the derivatives of the free energy. We demonstrate this approach on two different systems, the analytic van der Waals EOS and published data for the Lennard-Jones fluid, and we show that it is advantageous over direct learning of thermodynamic properties (i.e., not as derivatives of the free energy but as independent properties), in terms of both accuracy and the exact preservation of the Maxwell relations. Furthermore, the method implicitly provides the free energy of a system without explicit integration.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2023

IDAES-PSE 2.0 Release

The Institute for the Design of Advanced Energy Systems (IDAES) Integrated Platform is a versatile computational environment offering extensive process systems engineering (PSE) capabilities for optimizing the design and operation of complex, interacting technologies and systems. IDAES enables users to efficiently search vast, complex design spaces to discover the lowest cost, most environmentally sustainable solutions while supporting the full process modeling lifecycle, from conceptual design to dynamic optimization and control. The extensible, open platform empowers users to create models of novel processes and rapidly develop custom analyses, workflows, and end-user applications. IDAES-PSE 2.0.0 Release Highlights Removal of deprecated features from IDAES v1 Update to Pyomo v6.5 – this required a number of updates to support the new NL solver writer and to address some changes in Pyomo Creation of new testing suite for backward compatibility, model robustness and verification More general implementation of the Helmholtz EoS. This brings some new features like standard property diagrams, choice of mass or mole basis, and new state variable options Standardizing names in Heat Exchanger models (breaking change from v2.0.0a2): Control Volumes named hot_side and cold_side Ports names hot_side_inlet, hot_side_outlet, cold_side_inlet and cold_side_outlet Config Blocks names hot_side_config and cold_side_config Config arguments for user provided names for each side: hot_side_name and cold_side_name. Updating Keras surrogate tool to use v1.1 of OMLT New prototype API for model initialization (idaes.core.initialization) The new API uses "Model Initializer" objects instead of class methods, allowing for the definition of multiple initialization routines for a single model A number of common, model agnostic initialization routines have also been defined, including initialization from data, block-decomposition and a general hierarchical approach equivalent to the existing method for common unit models New metadata for thermophysical properties – valid_range This can be used to record the range of values over which a property value can be trusted, such as the range of experimental data used to regress parameters A number of new utility functions have been added to check for properties with values outside the valid range and to set bounds based on this metadata Updated construction of balance expressions in Control Volumes to remove unneeded terms In the past, unneeded terms were added as a constant 0 term, however they will now be dropped entirely from the expression This was necessary due to more strict unit checking in the new Pyomo solver writer which no longer ignores 0 terms Updates to metadata for thermophysical properties to better define known properties and units of measurement This results in more strict enforcement of standard naming for thermophysical and reaction properties Users can still define custom properties, but these must be done explicitly using the define_custom_properties() method instead of being implicitly created by add_property() Updated convergence tester utility tool to support definition of benchmark files (JSON format) and comparison of performance to benchmarks Set default iteration limit for IPOPT in IDAES config to 200 iterations Update scaling of example models to work with new Pyomo NL solver writer Improve testing of extensions and examples infrastructure to avoid need for downloading files Updated distillation column to centralize common functionality and remove a number of Pyomo warnings

IDAES↗