New directional vector limiters for discontinuous Galerkin methods

Hajduk, Hennes; Kuzmin, Dmitri; Aizinger, Vadym

doi:10.1016/j.jcp.2019.01.032

Cited by 18 publications

(12 citation statements)

References 32 publications

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“…Building on recent advances in the field of bound-preserving high-resolution finite element schemes for hyperbolic systems [13,18,20,22,44], the proposed upgrade of the FCT algorithm developed in [8] correction of the low-order predictor and sequential limiting subject to global energy constraints. The use of directional maximum principles for the momentum and magnetic field leads to less restrictive conditions than the imposition of local bounds on the kinetic and magnetic energy.…”

Section: Discussionmentioning

confidence: 99%

“…At the next step of the sequential limiting process, we constrain the momentum to satisfy the directional local maximum principles (cf. [22]) ρ e i u min i,k ≤ (ρ e iū e,L + α e,u k g e,ρu i ) • e k ≤ρ e i u max i,k ,…”

Section: Sequential a Priori Limitingmentioning

confidence: 99%

“…An upper bound α e,ρu k for correction factors satisfying conditions (74) for all nodes i ∈ N e can be determined similarly to α e,ρ and α e,ρE . For details and alternative choices of orthogonal limiting directions, we refer the reader to [22]. The limiter for the magnetic field is similar to that for the momentum.…”

Section: Sequential a Priori Limitingmentioning

confidence: 99%

“…After a priori limiting (as described in Sections 5.1-5.2) and a posteriori limiting (as described in Section 5.3), the localized element-based FCT algorithm produces nodal states U e,C i = [ρ e i , (ρu) e i , (ρE) e i , B e i ] T belonging to the subset G i ∩ G of the invariant set G defined by (25). Since the pressure p(U ) defined by the equation of state 4is a concave function of U for ρ > 0, substitution of U e,C i into (38) will produce a positivity-preserving convex average U C i ∈ G. This property can easily be shown using Jensen's inequality [13,22].…”

Section: Positivity Preservationmentioning

confidence: 99%

“…The calculation of correction factors for the synchronized FCT limiter is based solely on the density constraints. The sequential FCT algorithm is based on the methodology developed in [13,22] for continuous and discontinuous Galerkin discretizations of the Euler equations. Imposing local maximum principles on the density, velocity, specific total energy, and magnetic field, it uses different correction factors for different variables.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Kuzmin

Klyushnev

2020

Journal of Computational Physics

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This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks.

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Section: Discussionmentioning

confidence: 99%

Section: Sequential a Priori Limitingmentioning

confidence: 99%

Section: Sequential a Priori Limitingmentioning

confidence: 99%

Section: Positivity Preservationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Kuzmin

Klyushnev

2020

Journal of Computational Physics

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Shallow Water DG Simulations on FPGAs: Design and Comparison of a Novel Code Generation Pipeline

Alt,

Kenter,

Faghih-Naini

et al. 2023

Lecture Notes in Computer Science

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p-adaptive discontinuous Galerkin method for the shallow water equations with a parameter-free error indicator

Faghih-Naini

Aizinger

2022

Int J Geomath

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We propose a p-adaptive quadrature-free discontinuous Galerkin method for the shallow water equations based on a computationally efficient adaptivity indicator which works without any problem-dependent parameters. The error and smoothness of the solution are detected using the information collected for slope limiting and, for piecewise constant discretizations, by carrying out a reconstruction procedure. The accuracy and robustness of the new scheme are evaluated using several benchmarks and compared to other adaptivity indicators. Our results indicate that the proposed indicator finds a good balance between solution quality and computational overhead.

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New directional vector limiters for discontinuous Galerkin methods

Cited by 18 publications

References 32 publications

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Limiting and divergence cleaning for continuous finite element discretizations of the MHD equations

Shallow Water DG Simulations on FPGAs: Design and Comparison of a Novel Code Generation Pipeline

p-adaptive discontinuous Galerkin method for the shallow water equations with a parameter-free error indicator

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