Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

Mabuza, Sibusiso; Shadid, John N.; Kuzmin, Dmitri

doi:10.1016/j.jcp.2018.01.048

Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

Journal Article · Fri Feb 02 00:00:00 EST 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.01.048· OSTI ID:1464187

Mabuza, Sibusiso ^[1]; Shadid, John N. ^[2]; Kuzmin, Dmitri ^[3]

Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research; Univ. of New Mexico, Albuquerque, NM (United States). Dept. of Mathematics and Statistics
Technical Univ. of Dortmund, Dortmund (Germany). Inst. of Applied Mathematics (LS III)

The objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous Galerkin (CG) discretization in space. A CG semi-discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing (LED) condition for systems is used to discretize a system of conservation equations in space. The low order artificial diffusion is based on approximate Riemann solvers for hyperbolic conservation laws. In this case we consider both Rusanov and Roe artificial diffusion operators. In the Rusanov case, two designs are considered, a nodal based diffusion operator and a local projection stabilization operator. The result is a discretization that is LED and has first order convergence behavior. To achieve high resolution, limited antidiffusion is added back to the semi-discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator. The procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms. To further deal with phase errors (or terracing) common in FCT type methods, high order background dissipation is added to the antidiffusive correction. The resulting stabilized semi-discrete scheme can be discretized in time using a wide variety of time integrators. Numerical examples involving nonlinear scalar Burgers equation, and several shock hydrodynamics simulations for the Euler system are considered to demonstrate the performance of the method. In conclusion, for time discretization, Crank–Nicolson scheme and backward Euler scheme are utilized.

View Accepted Manuscript (DOE)

Research Organization:: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC04-94AL85000; NA0003525

OSTI ID:: 1464187

Alternate ID(s):: OSTI ID: 1548780

Report Number(s):: SAND--2017-8659J; 663067

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 361; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Monotonicity-preserving finite element schemes with adaptive mesh refinement for hyperbolic problems Bonilla, Jesús; Badia, Santiago Journal of Computational Physics, Vol. 416 https://doi.org/10.1016/j.jcp.2020.109522	journal	September 2020
On differentiable local bounds preserving stabilization for Euler equations Badia, Santiago; Bonilla, Jesús; Mabuza, Sibusiso arXiv https://doi.org/10.48550/arxiv.1911.06792	text	January 2019

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