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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Electromagnetic full-$f$ gyrokinetics in the tokamak edge with discontinuous Galerkin methods

Here, we present an energy-conserving discontinuous Galerkin scheme for the full-f electromagnetic gyrokinetic system in the long-wavelength limit. We use the symplectic formulation and solve directly for ∂A II /∂t, the inductive component of the parallel electric field, using a generalized Ohm’s law derived directly from the gyrokinetic equation. Linear benchmarks are performed to verify the implementation and show that the scheme avoids the Amp`ere cancellation problem. We perform a nonlinear electromagnetic simulation in a helical open-field-line system as a rough model of the tokamak scrape-off layer using parameters from the National Spherical Torus Experiment (NSTX). This is the first published nonlinear electromagnetic gyrokinetic simulation on open field lines. Comparisons are made to a corresponding electrostatic simulation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. Here we use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

thornado-hydro: A Discontinuous Galerkin Method for Supernova Hydrodynamics with Nuclear Equations of State

This paper describes algorithms for non-relativistic hydrodynamics in the toolkit for high-order neutrino radiation hydrodynamics (thornado), which is being developed for multiphysics simulations of core-collapse supernovae (CCSNe) and related problems with Runge–Kutta discontinuous Galerkin (RKDG) methods. More specifically, thornado employs a spectral type nodal collocation approximation, and we have extended limiters — a slope limiter to prevent non-physical oscillations and a bound-enforcing limiter to prevent non-physical states — from the standard RKDG framework to be able to accommodate a tabulated nuclear equation of state (EoS). To demonstrate the efficacy of the algorithms with a nuclear EoS, we first present numerical results from basic test problems in idealized settings in one and two spatial dimensions, employing Cartesian, spherical-polar, and cylindrical coordinates. Then, we apply the RKDG method to the problem of adiabatic collapse, shock formation, and shock propagation in spherical symmetry, initiated with a 15 M ⊙ progenitor. Herein, we find that the extended limiters improve the fidelity and robustness of the RKDG method in idealized settings. The bound-enforcing limiter improves robustness of the RKDG method in the adiabatic collapse application, while we find that slope limiting in characteristic fields is vulnerable to structures in the EoS — more specifically, in the phase transition from nuclei and nucleons to bulk nuclear matter. The success of these applications marks an important step toward applying RKDG methods to more realistic CCSN simulations with thornado in the future.

Nuclear astrophysics (1129)↗
DOE OSTI2025

Realizability-preserving discontinuous Galerkin method for spectral two-moment radiation transport in special relativity

Here we present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-moment model. Using a maximum-entropy closure, we solve for the Eulerian-frame energy and momentum. The proposed numerical method is designed to preserve moment realizability, which corresponds to moments defined by a nonnegative phase-space density. The realizability-preserving method is achieved with the following key components: (i) a discontinuous Galerkin phase-space discretization with specially constructed numerical fluxes in the spatial and energy dimensions; (ii) a strong stability-preserving implicit-explicit time-integration method; (iii) a realizability-preserving conserved to primitive moment solver; (iv) a realizability-preserving implicit collision solver; and (v) a realizability-enforcing limiter. Component (iii) is necessitated by the closure procedure, which closes higher order moments nonlinearly in terms of primitive moments. The nonlinear conserved to primitive and the implicit collision solves are formulated as fixed-point problems, which are solved with custom iterative solvers designed to preserve the realizability of each iterate. With a series of numerical tests, we demonstrate the accuracy and robustness of this discontinuous-Galerkin-implicit-explicit method.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRSDate not supplied

A Moving Discontinuous Galerkin Method with Interface Condition Enforcement for Reacting Hypersonic Flows

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced, is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this talk. Moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) [1],[2] are then presented for solving compressible flow problems with discontinuities based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. In the MDG-ICE formulation, both conservative quantities and grid geometry are considered as independent variables. A space-time DG formulation is used to solve the multi-material compressible Euler equations in the standard discontinuous solution space and the discrete grid geometry is solved using a variational formulation in a continuous space. A self-adaptive Levenberg-Marquardt method is utilized to solve the resulting over-determined system of nonlinear equations arising from the MDG-ICE formulation. A number of numerical experiments for a variety of flow problems are conducted to assess the accuracy and performance of the MDG-ICE method. Numerical results obtained indicate that the MDG-ICE method is able to deliver the designed order of both h- and p-convergence even for discontinuous solutions, and detect all types of interfaces, via interface condition enforcement and satisfy, via grid movement, the compressible Euler equations and the associated interface condition.

Discontinuous Galekin↗
DOE OSTI2022

Block triangular preconditioning for stochastic Galerkin method

Here, in this paper, we develop a new block triangular preconditioner for solving partial differential equations with random coefficients. We prove spectral bounds for the preconditioned system. Several numerical examples are provided to demonstrate the efficiency of this preconditioner, especially for stochastic problems with large variance.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2001

Meshless Petrov-Galerkin Method Applied to Axisymmetric Problems

An axisymmetric Meshless Local Petrov-Galerkin (MLPG) algorithm is presented for the potential and elasticity problems. In this algorithm the trial and test functions are chosen from different spaces. By a judicious choice of these functions, the integrals involved in the weak form can be restricted to a local neighborhood. This makes the method truly meshless. The MLPG algorithm is used to study various potential and elasticity problems for which exact solutions are available. The sensitivity and effectiveness of the MLPG algorithm to various parameters such as the weight functions, basis functions and support domain radius, etc. was studied. The MLPG algorithm yielded accurate solutions for all weight functions, basis functions and support domain radii considered for all of the problems studied.

Raju, I. S.↗
DOE OSTI2020

Fourier Analyses of High-Order Continuous and Discontinuous Galerkin Methods

In this paper, we present a Fourier analysis of wave propagation problems subject to a class of continuous and discontinuous discretizations using high-degree Lagrange polynomials. This allows us to obtain explicit analytical formulas for the dispersion relation and group velocity and, for the first time to our knowledge, characterize analytically the emergence of gaps in the dispersion relation at specific wavenumbers, when they exist, and compute their specific locations. Wave packets with energy at these wavenumbers will fail to propagate correctly, leading to significant numerical dispersion. We also show that the Fourier analysis generates mathematical artifacts, and we explain how to remove them through a branch selection procedure conducted by analysis of eigenvectors and associated reconstructed solutions. The higher frequency eigenmodes, named erratic in this study, are also investigated analytically and numerically.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Efficient Low-Order Refined Preconditioners for High-Order Matrix-Free Continuous and Discontinuous Galerkin Methods

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on finite element method--spectral element method (FEM-SEM) equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order (p = 1) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in h and p. The method is extended to interior penalty and Bassi and Rebay (BR2) discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Finally, numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

In Dzanic (2024), a limiting approach for high-order discontinuous Galerkin schemes was introduced which allowed for imposing constraints on the solution continuously (i.e., everywhere within the element). While exact for linear constraint functionals, this approach only imposed a sufficient (but not the minimum necessary) amount of limiting for nonlinear constraint functionals. This short note shows how this limiting approach can be extended to allow exactness for general nonlinear quasiconcave constraint functionals through a nonlinear limiting procedure, reducing unnecessary numerical dissipation. Finally, some examples are shown for nonlinear pressure and entropy constraints in the compressible gas dynamics equations, where both analytic and iterative approaches are used.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with hp-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work, we extend to the hp-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh h, and the degree of the polynomial approximation p. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Furthermore, numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).

97 MATHEMATICS AND COMPUTING↗
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 NTRS2002

A Meshless Local Petrov-Galerkin Method for Euler-Bernoulli Beam Problems

An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions. The use of these functions shows that the weak form can be integrated with conventional Gaussian integration. The MLPG method was evaluated by applying the formulation to a variety of patch test and thin beam problems. The formulation successfully reproduced exact solutions to machine accuracy when test functions with C2 continuity and an appropriate order of basis functions are used.

Raju, I. S.↗
NASA NTRSDate not supplied

Explicit Discontinuous Galerkin Methods for Conservation Laws

The two explicit DG methods in this study are based on a ‘predictor-corrector’ formulation, the first introduced by Lörcher, Gassner, and Munz (2007, 2008) called space–time expansion discontinuous Galerkin or STE-DG scheme, and the second, introduced independently by the author (Huynh 2006, 2013) called the upwind moment scheme. The predictor step of the two methods is essentially identical using a Cauchy-Kovalevsky (CK) procedure, which involves no interaction of the data among neighboring cells. The corrector step also shares the same space-time integration formulation and is where interaction of the data among neighboring cells takes place; the difference, however, is in how the resulting space-time volume integral is estimated. As a consequence of the different estimates, for the case of advection in one spatial dimension (1D), the moment scheme has a CFL (Courant-Friedrichs-Lewy) condition of 1 for all p and is accurate to order 2p+1, i.e., it possesses the super accuracy property, whereas the STE-DG method has a more restrictive CFL condition and is accurate to the expected order of p+1. For 1D advection, compared with the CFL conditions of 1/(2p+1) of standard RK-DG (Runge-Kutta) scheme where space and time discretization are of the same order, the moment scheme allows a significantly larger time step size. It also turns out that the scheme yields a result identical to Van Leer’s scheme III (1977), which amounts to shifting the data a distance of advection corresponding to the time step and projecting the result onto the space of polynomial solutions. Contrary to Van Leer’s approach, however, the space-time ‘predictor-corrector’ formulation facilitates extensions to the case of systems of equations. Concerning 2D extensions, in the case of advection, when the flow is along the diagonal direction, the CFL conditions for the moment schemes become restrictive as will be shown by Fourier (Von Neumann) stability and accuracy analyses. Since the moment scheme employs the right Radau points as collocation points in time, the method is closely related to the implicit Radau IIA scheme, which is stable for any time step size. The role of Radau IIA in relieving stability restriction for these explicit DG schemes remains to be explored

Discontinuous Galerkin↗