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At least 73 records · Page 4
All sourcesNASA NTRSDOE OSTI
NASA NTRS2020

Multigrid Preconditioning for a Space-Time Spectral-Element Discontinuous-Galerkin Solver

In this work we examine a multigrid preconditioning approach in the context of a high- order tensor-product discontinuous-Galerkin spectral-element solver. We couple multigrid ideas together with memory lean and efficient tensor-product preconditioned matrix-free smoothers. Block ILU(0)-preconditioned GMRES smoothers are employed on the coarsest spaces. The performance is evaluated on nonlinear problems arising from unsteady scale- resolving solutions of the Navier-Stokes equations: separated low-Mach unsteady ow over an airfoil from laminar to turbulent ow. A reduction in the number of ne space iterations is observed, which proves the efficiency of the approach in terms of preconditioning the linear systems, however this gain was not reflected in the CPU time. Finally, the preconditioner is successfully applied to problems characterized by stiff source terms such as the set of RANS equations, where the simple tensor product preconditioner fails. Theoretical justification about the findings is reported and future work is outlined.

Franciolini, Matteo↗
NASA NTRS2014

DNS of Flows over Periodic Hills using a Discontinuous-Galerkin Spectral-Element Method

Direct numerical simulation (DNS) of turbulent compressible flows is performed using a higher-order space-time discontinuous-Galerkin finite-element method. The numerical scheme is validated by performing DNS of the evolution of the Taylor-Green vortex and turbulent flow in a channel. The higher-order method is shown to provide increased accuracy relative to low-order methods at a given number of degrees of freedom. The turbulent flow over a periodic array of hills in a channel is simulated at Reynolds number 10,595 using an 8th-order scheme in space and a 4th-order scheme in time. These results are validated against previous large eddy simulation (LES) results. A preliminary analysis provides insight into how these detailed simulations can be used to improve Reynoldsaveraged Navier-Stokes (RANS) modeling

DNS↗
NASA NTRS1999

A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

Larson, Mats G.↗
NASA NTRS2001

Discontinuous Galerkin Methods for NonLinear Differential Systems

This talk considers simplified finite element discretization techniques for first-order systems of conservation laws equipped with a convex (entropy) extension. Using newly developed techniques in entropy symmetrization theory, simplified forms of the discontinuous Galerkin (DG) finite element method have been developed and analyzed. The use of symmetrization variables yields numerical schemes which inherit global entropy stability properties of the PDE (partial differential equation) system. Central to the development of the simplified DG methods is the Eigenvalue Scaling Theorem which characterizes right symmetrizers of an arbitrary first-order hyperbolic system in terms of scaled eigenvectors of the corresponding flux Jacobian matrices. A constructive proof is provided for the Eigenvalue Scaling Theorem with detailed consideration given to the Euler equations of gas dynamics and extended conservation law systems derivable as moments of the Boltzmann equation. Using results from kinetic Boltzmann moment closure theory, we then derive and prove energy stability for several approximate DG fluxes which have practical and theoretical merit.

Barth, Timothy↗
NASA NTRS2015

De-Aliasing Through Over-Integration Applied to the Flux Reconstruction and Discontinuous Galerkin Methods

High-order methods are quickly becoming popular for turbulent flows as the amount of computer processing power increases. The flux reconstruction (FR) method presents a unifying framework for a wide class of high-order methods including discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV). It offers a simple, efficient, and easy way to implement nodal-based methods that are derived via the differential form of the governing equations. Whereas high-order methods have enjoyed recent success, they have been known to introduce numerical instabilities due to polynomial aliasing when applied to under-resolved nonlinear problems. Aliasing errors have been extensively studied in reference to DG methods; however, their study regarding FR methods has mostly been limited to the selection of the nodal points used within each cell. Here, we extend some of the de-aliasing techniques used for DG methods, primarily over-integration, to the FR framework. Our results show that over-integration does remove aliasing errors but may not remove all instabilities caused by insufficient resolution (for FR as well as DG).

High-Order↗
NASA NTRS2004

Discontinuous Galerkin Finite Element Method for Parabolic Problems

In this paper, we develop a time and its corresponding spatial discretization scheme, based upon the assumption of a certain weak singularity of parallel ut(t) parallel Lz(omega) = parallel ut parallel2, for the discontinuous Galerkin finite element method for one-dimensional parabolic problems. Optimal convergence rates in both time and spatial variables are obtained. A discussion of automatic time-step control method is also included.

Kaneko, Hideaki↗
NASA NTRS2016

DNS of Low-Pressure Turbine Cascade Flows with Elevated Inflow Turbulence Using a Discontinuous-Galerkin Spectral-Element Method

Recent progress towards developing a new computational capability for accurate and efficient high-fidelity direct numerical simulation (DNS) and large-eddy simulation (LES) of turbomachinery is described. This capability is based on an entropy- stable Discontinuous-Galerkin spectral-element approach that extends to arbitrarily high orders of spatial and temporal accuracy, and is implemented in a computationally efficient manner on a modern high performance computer architecture. An inflow turbulence generation procedure based on a linear forcing approach has been incorporated in this framework and DNS conducted to study the effect of inflow turbulence on the suction- side separation bubble in low-pressure turbine (LPT) cascades. The T106 series of airfoil cascades in both lightly (T106A) and highly loaded (T106C) configurations at exit isentropic Reynolds numbers of 60,000 and 80,000, respectively, are considered. The numerical simulations are performed using 8th-order accurate spatial and 4th-order accurate temporal discretization. The changes in separation bubble topology due to elevated inflow turbulence is captured by the present method and the physical mechanisms leading to the changes are explained. The present results are in good agreement with prior numerical simulations but some expected discrepancies with the experimental data for the T106C case are noted and discussed.

Garai, Anirban↗
DOE OSTI2025

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)↗
NASA NTRS2009

A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion

We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases.

Huynh, H. T.↗
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 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↗
NASA NTRSDate not supplied

Discontinuous Galerkin and Related Methods for ODE

Starting from the standard integral formulation, the DG method is derived here in differential form. The key ingredient is a polynomial called the correction function, which helps ‘correct’ the discontinuous solution by approximating the jump and yields a continuous one. Under the right Radau quadrature, this continuous solution is identical to the solutions by the right Radau collocation and the continuous Galerkin (CG) methods. Next, the correction function facilitates the construction of the associated implicit Runge-Kutta schemes (IRK-DG). Different quadratures for DG result in different IRK-DG methods: left Radau quadrature in Radau IA, right Radau quadrature in Radau IIA or right Radau collocation, and Gauss quadrature in a method called DG-Gauss. The construction of IRK-DG clarifies the meaning and facilitates the proofs of various 𝐵(𝑝), 𝐶(𝜂), and 𝐷(𝜁) conditions for accuracy. The two consequences of these conditions are that all 𝑠-stage IRK-DG methods are accurate to order 2𝑠− 1, and the IRK-DG methods of Radau type are unique. Numerical examples showing the behavior of the DG solutions are provided. In all, the correction function plays a key role and helps establish the relations among the DG, IRK DG, collocation, and CG methods.

Numerical Methods for Ordinary Differential Equati↗
NASA NTRS2004

Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

Hou, Gene↗