## Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems

## Abstract

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 timemore »

- Authors:

- 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)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1464187

- Alternate Identifier(s):
- OSTI ID: 1548780

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

Journal ID: ISSN 0021-9991; 663067; TRN: US1902374

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

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 361; Journal Issue: C; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Local bounds preserving stabilization; Continuous Galerkin; Hyperbolic systems; Local extremum diminishing; Nodal variational stabilization

### Citation Formats

```
Mabuza, Sibusiso, Shadid, John N., and Kuzmin, Dmitri. Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.01.048.
```

```
Mabuza, Sibusiso, Shadid, John N., & Kuzmin, Dmitri. Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems. United States. doi:10.1016/j.jcp.2018.01.048.
```

```
Mabuza, Sibusiso, Shadid, John N., and Kuzmin, Dmitri. Fri .
"Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems". United States. doi:10.1016/j.jcp.2018.01.048. https://www.osti.gov/servlets/purl/1464187.
```

```
@article{osti_1464187,
```

title = {Local bounds preserving stabilization for continuous Galerkin discretization of hyperbolic systems},

author = {Mabuza, Sibusiso and Shadid, John N. and Kuzmin, Dmitri},

abstractNote = {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.},

doi = {10.1016/j.jcp.2018.01.048},

journal = {Journal of Computational Physics},

number = C,

volume = 361,

place = {United States},

year = {2018},

month = {2}

}

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