Direct Discontinuous Galerkin Methods for the Reacting, Multi-Species, 3D Compressible Navier-Stokes Equations
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Mathematics and Computing↗
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Results for “discontinuous Galerkin method”
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Generalized Korn’s inequalities for piecewise $H¹$ and $H²$ vector fields
The purpose of this paper is to construct a new class of discrete generalized Korn’s inequalities for piecewise $H^1$ vector fields and piecewise $H^2$ vector fields in three-dimensional space. The resulting Korn’s inequalities are different from the standard Korn’s inequalities, as they involve the trace-free symmetric gradient operator, in place of the usual symmetric gradient operator. Furthermore, it is anticipated that the new generalized Korn’s inequalities will be useful for the analysis of a broad range of finite element methods, including mixed finite element methods and discontinuous Galerkin methods.
97 MATHEMATICS AND COMPUTING↗
Agglomeration-based geometric multigrid solvers for compact discontinuous Galerkin discretizations on unstructured meshes
Here, we present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Physics-preserving enriched Galerkin method for a fully-coupled thermo-poroelasticity model
This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Here, several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method.
15 GEOTHERMAL ENERGY↗
Modeling detonation with CartaBlanca simulations
The accurate modeling of high explosive (HE) detonation and the con- sequent large solid deformation, failure, plastic flow, porosity growth, and shock wave propagation is important because simulations can capture spatial and temporal features that experimental diagnostics cannot capture. However, the simulation of the explosive event poses challenges to a computational scientist. These include the accurate modeling of ductile damage, crack formation, plastic deformation, as well as physical and nu- merical instabilities. The material response can be history-dependent and subject to large material deformation. Our research simulates the impact of a high explosive (Detasheet) onto a tantulum metal plate. We performed the simulations using CartaBlanca at different mesh resolutions. We decided the study would be impactful if we perform the simulations with the Material Point Method. Differences were observed at the different mesh resolutions in velocity and nodal stress magnitude, so increased mesh resolutions may be required. In addition, the Discontinuous Galerkin method would be needed to account for the large gas expansion.
97 MATHEMATICS AND COMPUTING↗
Half-closed discontinuous Galerkin discretisations
Here we introduce the concept of half-closed nodes for nodal discontinuous Galerkin (DG) discretisations. Unlike more commonly used closed nodes in DG, where on every element nodes are placed on all of its boundaries, half-closed nodes only require nodes to be placed on a subset of the element's boundaries. The effect of using different nodes on DG operator sparsity is studied and we find in particular for there to be no difference in the sparsity pattern of the Laplace operator whether closed or half-closed nodes are used. On quadrilateral/hexahedral elements we use the Gauss-Radau points as the half-closed nodes of choice, which we demonstrate is able to speed up DG operator assembly in addition to leverage previously known superconvergence results. We also discuss in this work some linear solver techniques commonly used for Finite Element or discontinuous Galerkin methods such as static condensation and block-based methods, and how they can be applied to half-closed DG discretisations.
97 MATHEMATICS AND COMPUTING↗
Dyn$\mathrm{AMO}$: Multi-agent reinforcement learning for dynamic anticipatory mesh optimization with applications to hyperbolic conservation laws
Here we introduce DynAMO, a reinforcement learning paradigm for Dynamic Anticipatory Mesh Optimization. Adaptive mesh refinement is an effective tool for optimizing computational cost and solution accuracy in numerical methods for partial differential equations. However, traditional adaptive mesh refinement approaches for time-dependent problems typically rely only on instantaneous error indicators to guide adaptivity. As a result, standard strategies often require frequent remeshing to maintain accuracy. In the DynAMO approach, multi-agent reinforcement learning is used to discover new local refinement policies that can anticipate and respond to future solution states by producing meshes that deliver more accurate solutions for longer time intervals. By applying DynAMO to discontinuous Galerkin methods for the linear advection and compressible Euler equations in two dimensions, we demonstrate that this new mesh refinement paradigm can outperform conventional threshold-based strategies while also generalizing to different mesh sizes, remeshing and simulation times, and initial conditions.
97 MATHEMATICS AND COMPUTING↗
Angular-spatial hp -adaptivity for radiative transfer with discontinuous Galerkin spectral element methods
Radiative transfer is important for many science and engineering applications, and numerical simulations of radiative transfer can be challenging. For instance, the radiation field is seven-dimensional – three spatial, two angular, one wavelength, and one temporal – and often features steep gradients. Therefore, memory usage is a key issue. To reduce memory, some past work has investigated the use of adaptive mesh refinement (AMR), typically for either the spatial or angular coordinate, and typically for only h -adaptivity. Here, we propose the use of AMR for the spatial and angular coordinates together, and the use of h - and p -adaptivity together as hp -AMR for the potential for further memory savings. We implemented the proposed method for several test cases in two spatial and one angular dimension, with the discontinuous Galerkin spectral element method. These test cases featured highly anisotropic angular radiation, with or without steep spatial gradients. Our primary findings from these test cases were: (1) Angular hp -adaptivity can deliver the radiation solution with the same accuracy as, and with much less computational memory than, uniform angular h - or p -refinements, or angular h -adaptivity alone. This is most obvious when the incoming radiation is highly anisotropic, in which case the savings can be orders of magnitude. (2) Full spatial-angular hp -adaptivity is more efficient in solution representation, compared to solely spatial or solely angular -adaptivity. This is most evident when steep gradients are present in both the spatial and angular distribution. These results suggest that adaptive spatial- hp angular-refinement may perform well in large-scale seven-dimensional applications.
Adaptive refinement↗
A discontinuous Galerkin spectral element method for compressible reacting flows
High-order methods have recently been shown to be an effective tool for high-fidelity flow computations like direct numerical simulations and large-eddy simulations because of their strong balance between accuracy and computational cost. In this work, a high-order discontinuous Galerkin spectral element method (DGSEM) is developed to solve the chemically reacting Navier-Stokes equations. To handle the disparate length and time scales associated with these equations, we develop a novel method which combines the spectral accuracy of the SEM with the flexibility of the DG approach. The framework, implemented in the spectral element code Nek5000, is well suited to capture turbulence in smooth regions of the flow, while maintaining numerical stability in the presence of shocks. An entropy-residual based artificial viscosity is added to smooth shocked regions of flow, and a positivity-preserving limiter is implemented to suppress non-physical oscillations. These enhancements support the numerical stability of the hydrodynamic sub-step, which is decoupled from the chemistry integration through a second-order operator splitting method. Here, a series of smooth and discontinuous validation cases are presented in increasing physical and computational complexity for both inviscid and viscous flows. In particular, simulations of canonical one-dimensional and two-dimensional detonations are performed, and the high-order numerical results are validated against available literature data. Additional validation studies are carried out for classical three-dimensional numerical simulations of incompressible and compressible turbulent flows.
Compressible reacting flows↗
A high‐order discontinuous Galerkin approach for physics‐based thermospheric modeling
Abstract The accurate prediction of aerodynamic drag on satellites orbiting in the upper atmosphere is critical to the operational success of modern space technologies, such as satellite‐based communication or navigation systems, which have become increasingly popular in the last few years due to the deployment of constellations of satellites in low‐Earth orbit. As a result, physics‐based models of the ionosphere and thermosphere have emerged as a necessary tool for the prediction of atmospheric outputs under highly variable space weather conditions. This paper proposes a high‐fidelity approach for physics‐based space weather modeling based on the solution of the Navier–Stokes equations using a high‐order discontinuous Galerkin method, combined with a matrix‐free strategy suitable for high‐performance computing on GPU architectures. The approach consists of a thermospheric model that describes a chemically frozen neutral atmosphere in nonhydrostatic equilibrium driven by the external excitation of the Sun. A novel set of variables is considered to treat the low densities present in the upper atmosphere and to accommodate the wide range of scales present in the problem. At the same time, and unlike most existing approaches, radial and angular directions are treated in a nonsegregated approach. The study presents a set of numerical examples that demonstrate the accuracy of the approximation and validate the current approach against observational data along a satellite orbit, including estimates of established empirical and physics‐based models of the ionosphere‐thermosphere system. Finally, a one‐dimensional radial derivation of the physics‐based model is presented and utilized for conducting a parametric study of the main thermal quantities under various solar conditions.
Engineering↗
A sweeping positivity-preserving high-order finite difference WENO scheme for Euler equations
We develop a simple, high-order, conservative and robust positivity-preserving sweeping procedure for the density and the nonlinear pressure function in the compressible Euler equations. Using the scaling limiter in Zhang and Shu (J Comput Phys 229:3091–3120, 2010), we obtain a non-trivial extension of the scalar sweeping technique in Liu et al. (J Sci Comput 73:1028–1071, 2017) for the positivity of pressure. The sweeping procedure developed in this paper is a post-processing technique, which can be applied to any concave functions of the conserved variables in hyperbolic conservation law systems. Thus, it has applications beyond the Euler equations. This procedure preserves positivity and conservation of physical quantities without destroying the accuracy of the underlying scheme. The algorithm works for general schemes including finite difference, finite volume and discontinuous Galerkin methods; however, in this paper we focus on finite difference weighted essentially non-oscillatory (WENO) methods. As a result, we provide numerical tests of the fifth-order finite difference WENO scheme to demonstrate the accuracy and robustness of the technique.
Compressible Euler equations↗
Origin of Pulsed Radio Emission from Magnetars
Extended periods of radio pulsations have been observed for six magnetars, displaying characteristics different from those of ordinary pulsars. In this Letter, we argue that radio emission is generated in a closed, twisted magnetic flux bundle originating near the magnetic pole and extending beyond 100 km from the magnetar. The electron–positron flow in the twisted bundle has to carry electric current and, at the same time, experiences a strong drag from the radiation field of the magnetar. This combination forces the plasma into a “radiatively locked” state with a sustained two-stream instability, generating radio emission. We demonstrate this mechanism using novel first-principles simulations that follow the plasma behavior by solving the relativistic Vlasov equation with the discontinuous Galerkin method. First, using one-dimensional simulations, we demonstrate how radiative drag induces the two-stream instability, sustaining turbulent electric fields. When extended to two dimensions, the system produces electromagnetic waves, including superluminal modes capable of escaping the magnetosphere. We measure their frequency and emitted power and incorporate the local simulation results into a global magnetospheric model. The model explains key features of the observed radio emission from magnetars: its appearance after an X-ray outburst, wide pulse profiles, luminosities ∼10 30 erg s −1 , and a broad range of frequencies extending up to ∼100 GHz.
Astronomical simulations↗
A High-Order Discontinuous Galerkin Spectral Element Method for Compressible Reacting Flows
High-order methods have recently been shown to be an effective tool for high-fidelity flow computations like direct numerical simulations and large eddy simulations due to their strong balance between accuracy and computational cost. In this work, a high-order discontinuous Galerkin spectral element method (DGSEM) is developed to solve the chemically reactive Euler equations encountered in high-speed combustion. To handle the disparate length and time scales associated with these equations, we develop a novel method which combines the spectral accuracy of the SEM with the flexibility of DG approach. Thus, the framework is well suited to capture turbulence in smooth regions of the flow, while maintaining numerical stability in the presence of shocks. The numerical method is implemented within the spectral element solver Nek5000. Validation cases are conducted for both non-reactive and reactive discontinuous flows to demonstrate the solver capability. In particular, canonical one-dimensional and two-dimensional detonation simulations are performed and the high-order numerical results are validated against available literature data.
computational fluid dynamics (CFD)↗
Concurrent two-way coupling of global and local models across internal boundaries with non-matching discretizations
Coupling local and global models enables efficient simulation of multiscale systems, where global models capture large-scale behavior and local models, with enhanced physics, resolve finer details over a smaller region. Here, this paper presents a mathematically consistent method for coupling physics-based models of varying fidelity across adjacent, non-overlapping subdomains, even when discretizations do not match at the immersed interdomain interfaces. Incompressible Navier-Stokes equations (NSE) constitute the global model while residual-based turbulence model serves as the local high-fidelity model. In addition, a scalar advection-diffusion equation that models the convection of an active scalar field is appended to the turbulence model in the local domain. This scalar field does not have its complement in the global model, giving rise to unequal number of equations at the immersed boundary between local and global models. Interdomain coupling terms are derived via the Variational Multiscale Discontinuous Galerkin (VMDG) method with new developments in scale representation and efficient fine-scale estimation. While transient laminar flows modeled with NSE in the global domain can be resolved with relatively coarse mesh, turbulent flow calculations in the local model require much finer spatial discretizations as well as smaller time-step for appropriately resolving the turbulent flow physics. The proposed framework also accommodates non-matching meshes at the immersed boundaries. Test problems in 2D and 3D numerically showcase the concurrent two-way coupling of unknown fields across the immersed boundaries. The 3D test presents a case with an unequal number of equations, where the scalar field represents the convection of contaminant concentration. This provides more detailed physics in the local region and highlights its application in climate modeling and atmospheric sciences.
Variational Multiscale Discontinuous Galerkin (VMD↗
Adaptive Model Reduction of High-Order Solutions of Compressible Flows via Optimal Transport
Not Available
High-order methods↗
LDRD Abbreviated report: High-Order General-Discrete-Ordinates Method Enabling Efficient Deterministic Transport in Hydrodynamic Simulations
Deterministic transport simulations for national-security and energy applications often operate in high-dimensional phase-space, where accuracy and cost both become major challenges. A common numerical artifact in such problems is the “ray-effect,” which appears as unphysical streaks. Beyond misinterpretation, these artifacts can contaminate tightly coupled physics, such as fluid dynamics, radiation-hydrodynamics, and laser-plasma interactions, eroding the predictive capability of entire multiphysics workflows. Our objective was to make high-dimension studies practical on modern hardware while mitigating the ray-effect without relying on prohibitively expensive sampling approaches such as Monte Carlo methods. We developed the Generic Discretization Library (GenDiL), a Graphics Processing Unit (GPU)-first framework that uses high-order Discontinuous Galerkin (DG) methods and matrix-free algorithms to reduce memory usage and improve computational efficiency, critical for phase-space simulations. GenDiL supports phase-space adaptivity in both mesh size and polynomial order (hp-adaptivity) to place resolution only where it is needed. A central capability is Local Dimensional Refinement (LDR), which couples lower-dimension continuum models to higher-dimension kinetic models through stable and conservative interfaces, so that high-fidelity physics is applied only in regions where it is essential. Building on the GenDiL framework, we developed the General SN (GSN) family of algorithms as a true generalization of the polar SN approach (discrete ordinates, often denoted SN). Rather than tying discrete ordinates to a specific polar change of coordinates, GSN formulates transport on an arbitrary change of coordinates chosen to reduce ray-effect. We studied two complementary variants: an analytic variant, where the coordinate map is prescribed in advance by a closed-form function; and a data-driven variant, where a quantity of interest, such as the net flux, guides the coordinate system. GenDiL provides the library infrastructure for efficient GPU execution, but the GSN concept is algorithmic and independent of any one library. Across representative high-dimension tests, including non-symmetric solutions, both variants delivered strong ray-effect mitigation at practical cost, moving four- to six-dimensional analysis toward repeatable, routine studies.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Multiphysics and Multiscale Simulation Methods for Electromagnetic Energy Assisted Fossil Fuel to Hydrogen Conversion (Final Scientific/Technical Report)
This report summarizes the technical accomplishments of the four-year research project “Multiphysics and Multiscale Simulation Methods for Electromagnetic Energy Assisted Fossil Fuel to Hydrogen Conversion” (Award No. DE-FE0032092), conducted at Howard University and the University of Houston (subawardee) from September 2021 to August 2025. The project successfully achieved all four major objectives: 1. 3D Structural Characterization – Developed 3D optical imaging and mechanical sectioning methods to characterize catalyst distribution and support morphology in nickel foam substrates. Successfully reconstructed 3D geometries and imported them into COMSOL Multiphysics for electromagnetic simulations. 2. EM Hotspot Simulation – Created all-frequency stable electromagnetic formulations and 3D nodal discontinuous Galerkin (NDG) methods for coupled electromagnetic-thermal-fluid problems in multiscale catalytic media. Demonstrated stable solutions from DC to microwave frequencies. 3. Multiphysics Coupling – Developed multiscale simulation methods coupling FEM electromagnetic solvers with thermal transport equations. Reactive molecular dynamics (ReaxFF MD) simulations were performed to investigate catalytic reaction mechanisms at the atomistic level. Demonstrated electromagnetic-thermal co-simulation capabilities for porous catalyst structures. 4. System Optimization – Designed and optimized EM-assisted catalytic systems using nickel foam and carbon foam structures, demonstrating significant temperature increases due to microwave heating. Observed and characterized plasma generation in carbon fiber catalysts. Investigated multiple reaction chamber geometries for improved microwave energy deposition. The project produced significant scientific contributions including 15+ peer-reviewed publications, trained multiple Ph.D. students and undergraduate researchers, and advanced the understanding of microwave-assisted hydrogen production from fossil fuels.
08 HYDROGEN↗
A family of independent Variable Eddington Factor methods with efficient preconditioned iterative solvers
We present a family of discretizations for the Variable Eddington Factor (VEF) equations that have high-order accuracy on curved meshes and efficient preconditioned iterative solvers. The VEF discretizations are combined with the Discontinuous Galerkin transport discretization from to form effective high-order, linear transport methods. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems presented by Arnold et al. to the VEF equations. This framework is used to define analogs of the interior penalty, second method of Bassi and Rebay, minimal dissipation local Discontinuous Galerkin, and continuous finite element methods. The analysis of subspace correction preconditioners, which use a continuous operator to iteratively precondition the discontinuous discretization, is extended to the case of the non-symmetric VEF system. Numerical results demonstrate that the VEF discretizations have arbitrary-order accuracy on curved meshes, preserve the thick diffusion limit, and are effective on a proxy problem from thermal radiative transfer in both outer transport iterations and inner preconditioned linear solver iterations. We demonstrate that the VEF solution converges to the S N transport solution as the mesh is refined on both problems with smooth and non-smooth behavior in angle. Parallel performance studies show that the interior penalty VEF discretization's linear solve weak scales out to 1024 processors and strong scales well on a single node. Particular attention is paid to the parallel performance of the VEF algorithm when used in combination with a parallel block Jacobi transport sweep.
97 MATHEMATICS AND COMPUTING↗