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DOE OSTI2024

A collision-based hybrid method for the BGK equation

In this article, we apply the collision-based hybrid method introduced by Hauck and McClarren to the Boltzmann equation with the BGK operator and a hyperbolic scaling. An implicit treatment of the source term is used to handle stiffness associated with the BGK operator. Although it helps the numerical scheme become stable with a large time step size, it is still not obvious to achieve the desired order of accuracy due to the relationship between the size of the spatial cell and the mean free path. Without asymptotic preserving property, a very restricted grid size is required to resolve the mean free path, which is not practical. Our approaches are based on the noncollision-collision decomposition of the BGK equation. We introduce the arbitrary order of nodal discontinuous Galerkin (DG) discretization in space with a semi-implicit time-stepping method; we employ the backward Euler time integration for the uncollided equation and the 2nd order predictor-corrector scheme for the collided equation, i.e., both source terms in uncollided and collided equations are treated implicitly and only streaming term in the collided equation is solved explicitly. This improves the computational efficiency without the complexity of the numerical implementation. Numerical results are presented for various Knudsen numbers to present the effectiveness and accuracy of our hybrid method. Also, we compare the solutions of the hybrid and non-hybrid schemes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A quadratic programming flux correction method for high-order DG discretizations of S transport

In this work, we present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the S N transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle. There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ~31% greater than performing one without fixup. The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Low-Order Preconditioning for the High-Order Finite Element de Rham Complex

Here, we present a unified framework for constructing spectrally equivalent low-order-refined discretizations for the high-order finite element de Rham complex. This theory covers diffusion problems in H 1 , H(curl), and H(div) and is based on combining a low-order discretization posed on a refined mesh with a high-order basis for Nédélec and Raviart–Thomas elements that makes use of the concept of polynomial histopolation (polynomial fitting using prescribed mean values over certain regions). This spectral equivalence, coupled with algebraic multigrid methods constructed using the low-order discretization, results in highly scalable matrix-free preconditioners for high-order finite element problems in the full de Rham complex. Additionally, a new lowest-order (piecewise constant) preconditioner is developed for high-order interior penalty discontinuous Galerkin (DG) discretizations, for which spectral equivalence results and convergence proofs for algebraic multigrid methods are provided. In all cases, the spectral equivalence results are independent of polynomial degree and mesh size; for DG methods, they are also independent of the penalty parameter. These new solvers are flexible and easy to use; any “black-box” preconditioner for low-order problems can be used to create an effective and efficient preconditioner for the corresponding high-order problem. A number of numerical experiments are presented, based on an implementation in the finite element library MFEM. A range of challenging three-dimensional problems are used to corroborate the theoretical properties and demonstrate the flexibility and scalability of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Automatically adaptive stabilized finite elements and continuation analysis for compaction banding in geomaterials

Under compressive creep, viscoplastic solids experiencing internal mass transfer processes can accommodate singular cnoidal wave solutions as material instabilities at the stationary wave limit. These instabilities appear when the loading rate is significantly faster than the material's capacity to diffusive internal perturbations, leading to localized failure features (e.g., cracks and compaction bands). These cnoidal waves, generally found in fluids, have strong nonlinearities that produce periodic patterns. Due to the singular nature of the solutions, the applicability of the theory is currently limited. Additionally, practical simulation tools require proper regularization to overcome the challenges that singularity induces. We focus on the numerical treatment of the governing equation using a nonlinear approach building on a recent adaptive stabilized finite element method. This automatic refinement method provides an error estimate that drives mesh adaptivity, a crucial feature for the problem at hand. We compare the performance of this adaptive strategy against analytical and standard finite element solutions. We then investigate the sensitivity of the diffusivity ratio, the parameter controlling the process, and identify multiple possible solutions with several stress peaks. We also show the evolution of the spacing between peaks for all solutions as a function of that parameter.

42 ENGINEERING↗
DOE OSTI2025

An Accurate SUPG-stabilized Continuous Galerkin Discretization for Anisotropic Heat Flux in Magnetic Confinement Fusion

We present a novel spatial discretization for the anisotropic heat conduction equation, aimed at improved accuracy at the high levels of anisotropy seen in a magnetized plasma, for example, for magnetic confinement fusion. The new discretization is based on a mixed formulation, introducing a form of the directional derivative along the magnetic field as an auxiliary variable and discretizing both the temperature and auxiliary fields in a continuous Galerkin (CG) space. Both the temperature and auxiliary variable equations are stabilized using the streamline upwind Petrov–Galerkin (SUPG) method, ensuring a better representation of the directional derivatives and therefore an overall more accurate solution. This approach can be seen as the CG-based version of our previous work (Wimmer, Southworth, Gregory, Tang, 2024), where we considered a mixed discontinuous Galerkin (DG) spatial discretization including DG-upwind stabilization. We prove consistency of the novel discretization, and demonstrate its improved accuracy over existing CG-based methods in test cases relevant to magnetic confinement fusion. This includes a long-run tokamak equilibrium sustainment scenario, demonstrating a 35% and 32% spurious heat loss for existing primal and mixed CG-based formulations versus 4% for our novel SUPG-stabilized discretization.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

A Predictor-Corrector Strategy for Adaptivity in Dynamical Low-Rank Approximations

Here, in this paper, we present a predictor-corrector strategy for constructing rank-adaptive, dynamical low-rank approximations (DLRAs) of matrix-valued ODE systems. The strategy is a compromise between (i) low-rank step-truncation approaches that alternately evolve and compress solutions and (ii) strict DLRA approaches that augment the low-rank manifold using subspaces generated locally in time by the DLRA integrator. The strategy is based on an analysis of the error between a forward temporal update into the ambient full-rank space, which is typically computed in a step-truncation approach before recompressing, and the standard DLRA update, which is forced to live in a low-rank manifold. We use this error, without requiring its full-rank representation, to correct the DLRA solution. A key ingredient for maintaining a low-rank representation of the error is a randomized SVD, which introduces some degree of stochastic variability into the implementation. The strategy is formulated and implemented in the context of discontinuous Galerkin spatial discretizations of PDEs and applied to several versions of DLRA methods found in the literature as well as a new variant. Numerical experiments comparing the predictor-corrector strategy to other methods demonstrate robustness to overcome shortcomings of step truncation or strict DLRA approaches: The former may require more memory than is strictly needed, while the latter may miss transients solution features that cannot be recovered. The effect of randomization, tolerances, and other implementation parameters is also explored.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Conservative velocity mappings for discontinuous Galerkin kinetics

Continuum computational kinetic plasma models evolve the distribution function of a plasma species f s on a phase-space grid over time. In many problems of interest the distribution function has limited extent in velocity space; hence, using a uniform, highly refined mesh would be costly and slow. Nonuniform velocity grids can reduce the computational cost by placing more degrees of freedom where f s is appreciable and fewer where it is not. In this work we introduce a first-of-its kind discontinuous Galerkin approach to nonuniform velocity-space discretization using mapped velocity coordinates. This new method is presented in the context of a gyrokinetic model used to study magnetized plasmas. We create discretizations of collisionless and collisional terms using mappings in a way that exactly conserves particles and energy. Numerical tests of such properties are presented, and we show that this new discretization can reproduce earlier gyrokinetic simulations using grids with up to 6–60 times fewer cells and 22X-60X speed-ups depending on dimensionality, geometry and plasma parameters.

Discontinuous Galerkin↗
DOE OSTI2025

A moment-conserving discontinuous Galerkin representation of the relativistic Maxwellian distribution

Kinetic simulations of relativistic gases and plasmas are critical for understanding diverse astrophysical and terrestrial systems, but the accurate construction of the relativistic Maxwellian, the Maxwell–Jüttner distribution, on a discrete simulation grid is challenging. Difficulties arise from the finite velocity bounds of the domain, which may not capture the entire distribution function, as well as errors introduced by projecting the function onto a discrete grid. Here, we present a novel scheme for iteratively correcting the moments of the projected distribution applicable to all grid-based discretizations of the relativistic kinetic equation. In addition, we describe how to compute the needed nonlinear quantities, such as Lorentz boost factors, in a discontinuous Galerkin scheme through a combination of numerical quadrature and weak operations. The resulting method accurately captures the distribution function and ensures that the moments match the desired values to machine precision.

astrophysical plasmas↗
DOE OSTI2024

Train small, model big: Scalable physics simulators via reduced order modeling and domain decomposition

Numerous cutting-edge scientific technologies originate at the laboratory scale, but transitioning them to practical industry applications is a formidable challenge. Traditional pilot projects at intermediate scales are costly and time-consuming. An alternative, the pilot-scale model, relies on high-fidelity numerical simulations, but even these simulations can be computationally prohibitive at larger scales. To overcome these limitations, we propose a scalable, physics-constrained reduced order model (ROM) method. The ROM identifies critical physics modes from small-scale unit components, projecting governing equations onto these modes to create a reduced model that retains essential physics details. We also employ Discontinuous Galerkin Domain Decomposition (DG-DD) to apply ROM to unit components and interfaces, enabling the construction of large-scale global systems without data at such large scales. Here this method is demonstrated on the Poisson and Stokes flow equations, showing that it can solve equations about 15–40 times faster with only ~1% relative error. Furthermore, ROM takes one order of magnitude less memory than the full order model, enabling larger scale predictions at a given memory limitation.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

A Fast Algebraic Multigrid Solver and Accurate Discretization for Highly Anisotropic Heat Flux I: Open Field Lines

We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization’s superior accuracy over other discretizations of anisotropic heat flux, achieving error 1000x smaller for anisotropy ratio of 10 9 , for closed field lines. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Finite Element Method for Electrochemical Transport

This code provides Finite Element solvers for Electrochemical Transport. A few examples from the literature are reproduced, with a focus on CO2 electrolysis. The Discontinuous Galerkin scheme for the electroneutral Nernst-Planck equations is from Roy, T., Andrej, J. and Beck, V.A., 2021. A scalable DG solver for the electroneutral Nernst-Planck equations. arXiv preprint arXiv:2112.09271. This work also includes a scalable preconditioner.

Beck, VictorA↗
DOE OSTI2023

A finite element method for angular discretization of the radiation transport equation on spherical geodesic grids

Discrete ordinate (S N ) and filtered spherical harmonics (FP N ) based schemes have been proven to be robust and accurate in solving the Boltzmann transport equation but they have their own strengths and weaknesses in different physical scenarios. We present a new method based on a finite element approach in angle that combines the strengths of both methods and mitigates their disadvantages. The angular variables are specified on a spherical geodesic grid with functions on the sphere being represented using a finite element basis. A positivity-preserving limiting strategy is employed to prevent non-physical values from appearing in the solutions. Here, the resulting method is then compared with both S N and FP N schemes using four test problems and is found to perform well when one of the other methods fail.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Final Technical Report for the SciDAC Center for the Study of Plasma Microturbulence

This final technical report for the SciDAC Center for the Study of Plasma Microturbulence documents contributions from MIT to the project over the period of the award (February 15, 2011 to February 14, 2020). This project focused on the development of neoclassical and gyrokinetic simulation capabilities relevant for the description of particle, momentum and energy transport in tokamak fusion plasmas. Extensive validation against experimental results from Alcator C-Mod and DIII-D experiments was performed. Overall areas of emphasis included gyrokinetic simulations, experiments, and observation of density gradient driven Trapped Electron Mode (TEM) turbulence relevant to burning plasmas; theory and gyrokinetic simulations of the parametric variation of the nonlinear upshift of the TEM critical density gradient; identification of a new mechanism for the isotope effect due to density gradient driven TEM turbulence; new synthetic diagnostics based on full-wave simulations which enabled unprecedented agreement between gyrokinetic simulations and measured fluctuation spectra with and without strong electron heating; development of new continuum pedestal neoclassical and turbulence codes; formulation and implementation of the first gyrokinetic exact Fokker-Planck collision operator in gyrokinetic turbulence simulations; development of new spectral approaches in velocity space for highly efficient treatments of collisions; new model collision operators implemented in the discontinuous Galerkin gyrokinetic edge turbulence code GKEYLL; and a new model and spectral code to study the interaction of ion- and electron-scale turbulence, which will serve as an efficient test-bed for multi-rate and multi-scale methods. These fundamental advances contribute to the development of a validated, predictive capability for turbulent and collisional particle, momentum and energy transport in both the core and edge of magnetically confined fusion plasmas.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Solution Irregularity Remediation for Spatial Discretization Error Estimation for S N Transport Solutions

The discrete ordinates linear Boltzmann transport equation is typically solved in its spatially discretized form, incurring spatial discretization error. Quantification of this error for purposes such as adaptive mesh refinement or error analysis requires an a posteriori estimator, which utilizes the numerical solution to the spatially discretized equation to compute an estimate. Because the quality of the numerical solution informs the error estimate, irregularities, present in the true solution for any realistic problem configuration, tend to cause the largest deviation in the error estimate vis-a-vis the true error. In this paper, an analytical partial singular characteristic tracking (pSCT) procedure for reducing the estimator’s error is implemented within our novel residual source estimator for a zeroth-order discontinuous Galerkin scheme, at the additional cost of a single inner iteration. Here, a metric-based evaluation of the pSCT scheme versus the standard residual source estimator is performed over the parameter range of a Method of Manufactured Solutions test suite. The pSCT scheme generates near-ideal accuracy in the estimate in problems where the dominant source of the estimator’s error is the solution irregularity, namely, problems where the true solution is discontinuous and problems where the true solution’s first derivative is discontinuous and the scattering ratio is low. In problems where the scattering ratio is high and the true solution is discontinuous in the first derivative, the error in the scattering source, which is not converged by the pSCT scheme, is greater than the error incurred due to the irregularity. Ultimately, a pSCT scheme is judged to be useful for error estimation in problems where the computational cost of the scheme is justified. In the presence of many irregularities, such a scheme may be intractable for general use, but in benchmarks, as an analytical tool, or in problems that have nondissipative discontinuities, the scheme may prove invaluable.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

A numerical study of particle jetting in a dense particle bed driven by an air-blast

In this work, the particle jetting behavior in a blast-driven dense particle bed is studied at early times. Four-way coupled Euler–Lagrange simulations are performed using a high-order discontinuous Galerkin spectral element solver coupled with a high-order Lagrangian particle solver, wherein the inter-particle collisions are resolved using a discrete element method collision model. Following the experiments of Rodriguez et al. [“Formation of particle jetting in a cylindrical shock tube,” Shock Waves 23(6), 619–634 (2013)] and the simulations of Osnes et al. [“Numerical simulation of particle jet formation induced by shock wave acceleration in a Hele-Shaw cell,” Shock Waves 28(3), 451–461 (2018)], the simulations are performed in a quasi-two-dimensional cylindrical geometry (Hele-Shaw cell). Parametric studies are carried out to assess the impact of the coefficient of restitution and the strength of the incident shock on the particle jetting behavior. The deposition of vorticity through a multiphase (gas–particle) analog of Richtmyer–Meshkov instability is observed to play a crucial role in channeling the particles into well-defined jets at the outer edge of the particle bed. This is confirmed by the presence of vortex pairs around the outer jets. Furthermore, the effect of the relaxation of the relative velocity between the two phases on the vorticity generation is explored by analyzing the correlation between the radial velocity of particles and the radial velocity of the gas at the particle location.

42 ENGINEERING↗
DOE OSTI2020

Regression Based Approach for Robust Finite Element Analysis on Arbitrary Grids. LDRD Final Report

This report summarizes the work performed under a one-year LDRD project aiming to enable accurate and robust numerical simulation of partial differential equations for meshes that are of poor quality. Traditional finite element methods use the mesh to both discretize the geometric domain and to define the finite element shape functions. The latter creates a dependence between the quality of the mesh and the properties of the finite element basis that may adversely affect the accuracy of the discretized problem. In this project, we propose a new approach for defining finite element shape functions that breaks this dependence and separates mesh quality from the discretization quality. At the core of the approach is a meshless definition of the shape functions, which limits the purpose of the mesh to representing the geometric domain and integrating the basis functions without having any role in their approximation quality. The resulting non-conforming space can be utilized within a standard discontinuous Galerkin framework providing a rigorous foundation for solving partial differential equations on low-quality meshes. We present a collection of numerical experiments demonstrating our approach in a wide range of settings: strongly coercive elliptic problems, linear elasticity in the compressible regime, and the stationary Stokes problem. We demonstrate convergence for all problems and stability for element pairs for problems which usually require inf-sup compatibility for conforming methods, also referring to a minor modification possible through the symmetric interior penalty Galerkin framework for stabilizing element pairs that would otherwise be traditionally unstable. Mesh robustness is particularly critical for elasticity, and we provide an example that our approach provides a greater than 5x improvement in accuracy and allows for taking an 8x larger stable timestep for a highly deformed mesh, compared to the continuous Galerkin finite element method. The report concludes with a brief summary of ongoing projects and collaborations that utilize or extend the products of this work.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multigrid strategies. For both approaches, we start from an integral formulation of the continuous time dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multi-grid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies are compared for varying orders of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences in execution times

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Entropy–Preserving and Entropy–Stable Relaxation IMEX and Multirate Time–Stepping Methods

In this work, we propose entropy-preserving and entropy-stable partitioned Runge–Kutta (RK) methods. In particular, we extend the explicit relaxation Runge–Kutta methods to IMEX–RK methods and a class of explicit second-order multirate methods for stiff problems arising from scale-separable or grid-induced stiffness in a system. The proposed approaches not only mitigate system stiffness but also fully support entropy-preserving and entropy-stability properties at a discrete level. The key idea of the relaxation approach is to adjust the step completion with a relaxation parameter so that the time-adjusted solution satisfies the entropy condition at a discrete level. The relaxation parameter is computed by solving a scalar nonlinear equation at each timestep in general; however, as for a quadratic entropy function, we theoretically derive the explicit form of the relaxation parameter and numerically confirm that the relaxation parameter works the Burgers equation. Several numerical results for ordinary differential equations and the Burgers equation are presented to demonstrate the entropy-conserving/stable behavior of these methods. We also compare the relaxation approach and the incremental direction technique for the Burgers equation with and without a limiter in the presence of shocks.

97 MATHEMATICS AND COMPUTING↗