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DOE OSTI2011

Implicit-explicit evolution of single black holes

Numerical simulations of binary black holes - an important predictive tool for the detection of gravitational waves - are computationally expensive, especially for binaries with high mass ratios or with rapidly spinning constituent holes. Existing codes for evolving binary black holes rely on explicit time-stepping methods, for which the time-step size is limited by the smallest spatial scale through the Courant-Friedrichs-Lewy condition. Binary inspiral typically involves spatial scales (the spatial resolution required by a small or rapidly spinning hole) which are orders of magnitude smaller than the relevant (orbital, precession, and radiation-reaction) time scales characterizing the inspiral. Therefore, in explicit evolutions of binary black holes, the time-step size is typically orders of magnitude smaller than the relevant physical time scales. Implicit time-stepping methods allow for larger time steps, and they often reduce the total computational cost (without significant loss of accuracy) for problems dominated by spatial rather than temporal error, such as for binary-black-hole inspiral in corotating coordinates. However, fully implicit methods can be difficult to implement for nonlinear evolution systems like the Einstein equations. Therefore, in this paper we explore implicit-explicit (IMEX) methods and use them for the first time to evolve black-hole spacetimes. Specifically, as a first step toward IMEX evolution of a full binary-black-hole spacetime, we develop an IMEX algorithm for the generalized harmonic formulation of the Einstein equations and use this algorithm to evolve stationary and perturbed single-black-hole spacetimes. Numerical experiments explore the stability and computational efficiency of our method.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2011

A generalized implicit algorithm for multi-dimensional particle-in-cell simulations in Cartesian geometry

An implicit multi-dimensional particle-in-cell (PIC) code is developed to study the interaction of ultrashort pulse lasers with matter. The algorithm is based on current density decomposition and is only marginally more complicated compared to explicit PIC codes, but it completely eliminates grid heating and possesses good energy conserving properties with relaxed time step and grid resolution. This is demonstrated in a test case study, in which high-energy protons are generated from a thin carbon foil at solid density using linear and circular polarizations. The grid heating rate is estimated to be 1-10 eV/ps.

36 MATERIALS SCIENCE↗

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 OSTI2024

FARM User Guidance and Instructions

The FARM (Feasible Actuator Range Modifier) module is a component of the RAVEN-based FORCE framework for the analysis of Integrated Energy Systems (IES). FARM aids HERON in the solution of the power dispatch problem by evaluating feasible set-point signals to be issued to the control systems of the different IES unit components. Set-points need to satisfy limits on both production variables (i.e., the variables to be optimized such as the electrical power, etc.) and process variables tied to the service life of equipment (e.g., steam flowrate, vessel pressure, turbine firing temperature, etc.). To enforce all these limits, a two-stage approach is adopted. First, the power dispatcher algorithm in HERON module estimates set-points that meet the constraints on the production variables, e.g., power levels and power ramp rate limits. These constraints are called explicit constraints. Then, if necessary, FARM adjusts these set-points to ensure the respect of the limits on the process variables of interest, given the knowledge of the system dynamics acquired through machine learning algorithms. These constraints are called implicit constraints. From this standpoint, FARM constitutes a bridge between the HERON power dispatcher that adopts a simplified description of the IES unit (low-resolution physics) and the HYBRID high-fidelity models (high-resolution physics). In this report, the general workflow and the software requirements of FARM module are summarized, and the detailed instructions for installing FARM software, running built-in example cases, deriving Linear Parameter-Varying (LPV) state-space models, and using FARM for user-defined power dispatch problems are provided.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2024

FARM supervisory capabilities for thermal energy storage

The FARM (Feasible Actuator Range Modifier) module is a component of the RAVEN-based FORCE framework for the analysis of Integrated Energy Systems (IES). FARM aids HERON in the solution of the power dispatch problem by evaluating feasible set-point signals to be issued to the control systems of the different IES unit components. Set-points need to satisfy limits on both production variables (i.e., the variables to be optimized such as the electrical power, etc.) and process variables tied to the service life of equipment (e.g., steam flowrate, vessel pressure, turbine firing temperature, etc.). To enforce all these limits, a two-stage approach is adopted. First, the power dispatcher algorithm in HERON module estimates set-points that meet the constraints on the production variables, e.g., power levels and power ramp rate limits. These constraints are called explicit constraints. Then, if necessary, FARM adjusts these set-points to ensure the respect of the limits on the process variables of interest, given the knowledge of the system dynamics acquired through machine learning algorithms. These constraints are called implicit constraints. From this standpoint, FARM constitutes a bridge between the HERON power dispatcher that adopts a simplified description of the IES unit (low-resolution physics) and the HYBRID high-fidelity models (high-resolution physics). In this report, an overview of the major capabilities of the latest release of FARM is provided, along with a summary of the tool demonstration campaign conducted at the Dynamic Energy Technology and Integration Laboratory (DETAIL) facility. These results assess the performance of the control system architecture embedding FARM both as a Validator of the HERON power dispatcher and as a real time Supervisory control scheme. Additionally, the report outlines the areas that FARM might benefit from, along with proposed solutions.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2018

A Comprehensive Comparison of Relativistic Particle Integrators

We compare relativistic particle integrators commonly used in plasma physics, showing several test cases relevant for astrophysics. Three explicit particle pushers are considered, namely, the Boris, Vay, and Higuera–Cary schemes. We also present a new relativistic fully implicit particle integrator that is energy conserving. Furthermore, a method based on the relativistic guiding center approximation is included. The algorithms are described such that they can be readily implemented in magnetohydrodynamics codes or Particle-in-Cell codes. Here, our comparison focuses on the strengths and key features of the particle integrators. We test the conservation of invariants of motion and the accuracy of particle drift dynamics in highly relativistic, mildly relativistic, and non-relativistic settings. The methods are compared in idealized test cases, i.e., without considering feedback onto the electrodynamic fields, collisions, pair creation, or radiation. The test cases include uniform electric and magnetic fields, ${\boldsymbol{E}}\times {\boldsymbol{B}}$ fields, force-free fields, and setups relevant for high-energy astrophysics, e.g., a magnetic mirror, a magnetic dipole, and a magnetic null. These tests have direct relevance for particle acceleration in shocks and in magnetic reconnection.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Time-explicit Darwin PIC algorithm

A new approach to Darwin particle-in-cell plasma simulation is described. Using a finite-element approach, the vector potential is assured to be exactly solenoidal. This allows writing the system of multi-species particles as an action-at-a-distance Hamiltonian system. Applying recently-developed explicit symplectic methods to this gives a time-explicit algorithm with all desired properties. Elliptic systems for the electrostatic and magneto-static portions of the problem are inverted efficiently by an algebraic multi-grid algorithm. The algorithm is implemented in a two-dimensional Cartesian-geometry code and tested by application to Weibel instability and to Alfvén wave dynamics. Results show the effectiveness of this approach. Extensions to arbitrary meshes and to a partially time-implicit scheme are briefly discussed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2012

REACTIVE TRANSPORT MODELING USING A PARALLEL FULLY-COUPLED SIMULATOR BASED ON PRECONDITIONED JACOBIAN-FREE NEWTON-KRYLOV

Systems of multicomponent reactive transport in porous media that are large, highly nonlinear, and tightly coupled due to complex nonlinear reactions and strong solution-media interactions are often described by a system of coupled nonlinear partial differential algebraic equations (PDAEs). A preconditioned Jacobian-Free Newton-Krylov (JFNK) solution approach is applied to solve the PDAEs in a fully coupled, fully implicit manner. The advantage of the JFNK method is that it avoids explicitly computing and storing the Jacobian matrix during Newton nonlinear iterations for computational efficiency considerations. This solution approach is also enhanced by physics-based blocking preconditioning and multigrid algorithm for efficient inversion of preconditioners. Based on the solution approach, we have developed a reactive transport simulator named RAT. Numerical results are presented to demonstrate the efficiency and massive scalability of the simulator for reactive transport problems involving strong solution-mineral interactions and fast kinetics. It has been applied to study the highly nonlinearly coupled reactive transport system of a promising in situ environmental remediation that involves urea hydrolysis and calcium carbonate precipitation.

58 GEOSCIENCES↗

DOE OSTI2024

Feasible Actuator Range Modifier (FARM), a Tool Aiding the Solution of Unit Dispatch Problems for Advanced Energy Systems

Integrated energy systems (IESs) seek to minimize power generating costs in future power grids through the coupling of different energy technologies. To accommodate fluctuations in load demand due to the penetration of renewable energy sources, flexible operation capabilities must be fully exploited, and even power plants that are traditionally considered as base-load units need to be operated according to unconventional paradigms. Thermomechanical loads induced by frequent power adjustments can accelerate the wear and tear. If a unit is flexibly operated without respecting limits on materials, the risk of failures of expensive components will eventually increase, nullifying the additional profits ensured by flexible operation. In addition to the bounds on power variations (explicit constraints),the solution of the unit dispatch problem needs to meet the limits on the variation of key process variables, including temperature, pressure and flow rate (implicit constraints).The FARM (Feasible Actuator Range Modifier) module was developed to enable existing optimization algorithms to identify solutions to the unit dispatch problem that are both economically favorable and technologically sustainable. Thanks to the iterative dispatcher–validator scheme, FARM permits addressing all the imposed constraints without excessively increasing the computational costs. In this work, the algorithms constituting the module are described, and the performance was assessed by solving the unit dispatch problem for an IES composed of three units, i.e., balance of plant, gas turbine, and high-temperature steam electrolysis. Finally, the FARM module provides dedicated tools for visualizing the response of the constrained variables of interest during operational transients and a tool aiding the operator at making decisions. These techniques might represent the first step towards the deployment of an ecological interface design (EID) for IES units.

47 OTHER INSTRUMENTATION↗

DOE OSTI2013

CASTRO: A NEW COMPRESSIBLE ASTROPHYSICAL SOLVER. III. MULTIGROUP RADIATION HYDRODYNAMICS

We present a formulation for multigroup radiation hydrodynamics that is correct to order O(v/c) using the comoving-frame approach and the flux-limited diffusion approximation. We describe a numerical algorithm for solving the system, implemented in the compressible astrophysics code, CASTRO. CASTRO uses a Eulerian grid with block-structured adaptive mesh refinement based on a nested hierarchy of logically rectangular variable-sized grids with simultaneous refinement in both space and time. In our multigroup radiation solver, the system is split into three parts: one part that couples the radiation and fluid in a hyperbolic subsystem, another part that advects the radiation in frequency space, and a parabolic part that evolves radiation diffusion and source-sink terms. The hyperbolic subsystem and the frequency space advection are solved explicitly with high-order Godunov schemes, whereas the parabolic part is solved implicitly with a first-order backward Euler method. Our multigroup radiation solver works for both neutrino and photon radiation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

Conservative tightly-coupled simulations of stochastic multiscale systems

Multiphysics problems often involve components whose macroscopic dynamics is driven by microscopic random fluctuations. The fidelity of simulations of such systems depends on their ability to propagate these random fluctuations throughout a computational domain, including subdomains represented by deterministic solvers. When the constituent processes take place in nonoverlapping subdomains, system behavior can be modeled via a domain-decomposition approach that couples separate components at the interfaces between these subdomains. Its coupling algorithm has to maintain a stable and efficient numerical time integration even at high noise strength. We propose a conservative domain-decomposition algorithm in which tight coupling is achieved by employing either Picard's or Newton's iterative method. Coupled diffusion equations, one of which has a Gaussian white-noise source term, provide a computational testbed for analysis of these two coupling strategies. Fully-converged (“implicit”) coupling with Newton's method typically outperforms its Picard counterpart, especially at high noise levels. This is because the number of Newton iterations scales linearly with the amplitude of the Gaussian noise, while the number of Picard iterations can scale superlinearly. At large time intervals between two subsequent inter-solver communications, the solution error for single-iteration (“explicit”) Picard's coupling can be several orders of magnitude higher than that for implicit coupling. Increasing the explicit coupling's communication frequency reduces this difference, but the resulting increase in computational cost can make it less efficient than implicit coupling at similar levels of solution error, depending on the communication frequency of the latter and the noise strength. This trend carries over into higher dimensions, although at high noise strength explicit coupling may be the only computationally viable option.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2022

Modeling wave propagation in elastic solids via high-order accurate implicit-mesh discontinuous Galerkin methods

Here, a high-order accurate implicit-mesh discontinuous Galerkin framework for wave propagation in single-phase and bi-phase solids is presented. The framework belongs to the embedded-boundary techniques and its novelty regards the spatial discretization, which enables boundary and interface conditions to be enforced with high-order accuracy on curved embedded geometries. High-order accuracy is achieved via high-order quadrature rules for implicitly-defined domains and boundaries, whilst a cell-merging strategy addresses the presence of small cut cells. The framework is used to discretize the governing equations of elastodynamics, written using a first-order hyperbolic momentum-strain formulation, and an exact Riemann solver is employed to compute the numerical flux at the interface between dissimilar materials with general anisotropic properties. The space-discretized equations are then advanced in time using explicit high-order Runge–Kutta algorithms. Several two- and three-dimensional numerical tests including dynamic adaptive mesh refinement are presented to demonstrate the high-order accuracy and the capability of the method in the elastodynamic analysis of single- and bi-phases solids containing complex geometries.

42 ENGINEERING↗

DOE OSTI2013

Liquid and liquid–gas flows at all speeds

All speed flows and in particular low Mach number flow algorithms are addressed for the numerical approximation of the Kapila et al. [1] multiphase flow model. This model is valid for fluid mixtures evolving in mechanical equilibrium but out of temperature equilibrium and is efficient for material interfaces computation separating miscible and non-miscible fluids. In this context, the interface is considered as a numerically diffused zone, captured as well as all present waves (shocks, expansion waves). The same flow model can be used to solve cavitating and boiling flows [2]. Many applications occurring with liquid–gas interfaces and cavitating flows involve a very wide range of Mach number, from 10{sup −3} to supersonic (and even hypersonic) conditions with respect to the mixture sound speed. It is thus important to address numerical methods free of restrictions regarding the Mach number. To do this, a preconditioned Riemann solver is built and embedded into the Godunov explicit scheme. It is shown that this method converges to exact solutions but needs too small time steps to be efficient. An implicit version is then derived, first in one dimension and second in the frame of 2D unstructured meshes. Two-phase flow preconditioning is then addressed in the frame of the Saurel et al. [3] algorithm. Modifications of the preconditioned Riemann solver are needed and detailed. Convergence of both single phase and two-phase numerical solutions are demonstrated with the help of single phase and two-phase steady nozzle flow solutions. Last, the method is illustrated by the computation of real cavitating flows in Venturi nozzles. Vapour pocket size and instability frequencies are reproduced by the model and method without using any adjustable parameter.

42 ENGINEERING↗

DOE OSTI2017

Stochastic simulation of reaction-diffusion systems: A fluctuating-hydrodynamics approach

Here, we develop numerical methods for stochastic reaction-diffusion systems based on approaches used for fluctuating hydrodynamics (FHD). For hydrodynamic systems, the FHD formulation is formally described by stochastic partial differential equations (SPDEs). In the reaction-diffusion systems we consider, our model becomes similar to the reaction-diffusion master equation (RDME) description when our SPDEs are spatially discretized and reactions are modeled as a source term having Poisson fluctuations. However, unlike the RDME, which becomes prohibitively expensive for an increasing number of molecules, our FHD-based description naturally extends from the regime where fluctuations are strong, i.e., each mesoscopic cell has few (reactive) molecules, to regimes with moderate or weak fluctuations, and ultimately to the deterministic limit. By treating diffusion implicitly, we avoid the severe restriction on time step size that limits all methods based on explicit treatments of diffusion and construct numerical methods that are more efficient than RDME methods, without compromising accuracy. Guided by an analysis of the accuracy of the distribution of steady-state fluctuations for the linearized reaction-diffusion model, we construct several two-stage (predictor-corrector) schemes, where diffusion is treated using a stochastic Crank-Nicolson method, and reactions are handled by the stochastic simulation algorithm of Gillespie or a weakly second-order tau leaping method. We find that an implicit midpoint tau leaping scheme attains second-order weak accuracy in the linearized setting and gives an accurate and stable structure factor for a time step size of an order of magnitude larger than the hopping time scale of diffusing molecules. We study the numerical accuracy of our methods for the Schlögl reaction-diffusion model both in and out of thermodynamic equilibrium. We demonstrate and quantify the importance of thermodynamic fluctuations to the formation of a two-dimensional Turing-like pattern and examine the effect of fluctuations on three-dimensional chemical front propagation. Furthermore, by comparing stochastic simulations to deterministic reaction-diffusion simulations, we show that fluctuations accelerate pattern formation in spatially homogeneous systems and lead to a qualitatively different disordered pattern behind a traveling wave.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

A novel implicit hybrid machine learning model and its application for reinforcement learning

A novel methodology to develop implicit hybrid models is presented. PyTorch is used to integrate physics-based equations with machine learning models. Automatic differentiation of the hybrid model is leveraged to solve the implicit equations. Iterative solving enables gradient based updates to the machine learning model. The novel methodology is compared to an explicit hybrid approach on a continuously stirred tank reactor (CSTR). The novel method results in a lower modelling error. Both hybrid models effectively train with noisy data. To test the implicit hybrid model, it is employed as a reinforcement learning (RL) training model. The RL algorithm trained on the hybrid model outperforms real time optimization of the CSTR and performs nearly as well as RL trained directly on the CSTR and a traditional gradient based approach. Training RL directly on the CSTR requires over 60,000 system interactions compared to 6000 historical data points for hybrid model development.

42 ENGINEERING↗

DOE OSTI2023

MrHyDE v.1.0

SAND2024-01324O MrHyDE, which stands for Multi-resolution Hybridized Differential Equations, is a general-purpose C++ package for the solution of coupled multiphysics and multiscale systems on massively parallel computing systems. MrHyDE is designed to enable moving beyond forward simulation for multiscale applications which includes optimization, control, uncertainty quantification, and stochastic inversion. The framework provides interfaces to several packages within the Trilinos framework and leverages automatic differentiation to enable adjoint capabilities for large-scale, gradient-based optimization. MrHyDE provides automated multiscale capabilities through a subgrid model interface and multiscale Dirichlet-to-Neumann maps. For extreme-scale applications, MrHyDE provides in situ data-compression algorithms to reduce memory requirements while maintaining performance. MrHyDE is a general-purpose, computational framework for the solution of multiscale and multiphysics applications. It uses a combination of structure-preserving, physics-compatible discretizations, fully implicit methods, multi-resolution schemes, or fully explicit methods. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

SciDAC↗

DOE OSTI2013

A Parallel, Fully Coupled, Fully Implicit Solution to Reactive Transport in Porous Media Using the Preconditioned Jacobian-Free Newton-Krylov Method

Modeling large multicomponent reactive transport systems in porous media is particularly challenging when the governing partial differential algebraic equations (PDAEs) are highly nonlinear and tightly coupled due to complex nonlinear reactions and strong solution-media interactions. Here we present a preconditioned Jacobian-Free Newton-Krylov (JFNK) solution approach to solve the governing PDAEs in a fully coupled and fully implicit manner. A well-known advantage of the JFNK method is that it does not require explicitly computing and storing the Jacobian matrix during Newton nonlinear iterations. Our approach further enhances the JFNK method by utilizing physics-based, block preconditioning and a multigrid algorithm for efficient inversion of the preconditioner. This preconditioning strategy accounts for self- and optionally, cross-coupling between primary variables using diagonal and off-diagonal blocks of an approximate Jacobian, respectively. Numerical results are presented demonstrating the efficiency and massive scalability of the solution strategy for reactive transport problems involving strong solution-mineral interactions and fast kinetics. We found that the physics-based, block preconditioner significantly decreases the number of linear iterations, directly reducing computational cost; and the strongly scalable algebraic multigrid algorithm for approximate inversion of the preconditioner leads to excellent parallel scaling performance.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Performance analysis and implementation details of the Energy Conserving Semi-Implicit Method code (ECsim)

We present in this work the implementation of the Energy Conserving Semi-Implicit Method in a parallel code called ECsim. This new code is a three-dimensional, fully electromagnetic particle in cell (PIC) code. It is written in C/C++ and uses MPI to allow massive parallelization. ECsim is unconditionally stable in time, eliminates the finite grid instability, and has the same cycle scheme as the explicit code with a computational cost comparable to other semi-implicit PIC codes. All these features make it a very valuable tool to address situations which have not been possible to analyze until now with other PIC codes. In this work, we show the details of the algorithm implementation and we study its performance in different systems. ECsim is compared with another semi-implicit PIC code with different time and spectral resolution, showing its ability to address situations where other codes fail.

97 MATHEMATICS AND COMPUTING↗