## Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates

## Abstract

In this paper, we present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface ofmore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA); LANL Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1459834

- Alternate Identifier(s):
- OSTI ID: 1532565

- Report Number(s):
- LA-UR-18-21449

Journal ID: ISSN 0021-9991; TRN: US1901810

- Grant/Contract Number:
- AC52-06NA25396; LA-UR-17-30310

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 373; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Lagrangian hydrodynamics; discontinuous Galerkin; cell-centered hydrodynamics; axisymmetric coordinates; true volume approach; area-weighted approach

### Citation Formats

```
Liu, Xiaodong, Morgan, Nathaniel R., and Burton, Donald E. Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates. United States: N. p., 2018.
Web. doi:10.1016/j.jcp.2018.06.073.
```

```
Liu, Xiaodong, Morgan, Nathaniel R., & Burton, Donald E. Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates. United States. doi:10.1016/j.jcp.2018.06.073.
```

```
Liu, Xiaodong, Morgan, Nathaniel R., and Burton, Donald E. Tue .
"Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates". United States. doi:10.1016/j.jcp.2018.06.073. https://www.osti.gov/servlets/purl/1459834.
```

```
@article{osti_1459834,
```

title = {Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates},

author = {Liu, Xiaodong and Morgan, Nathaniel R. and Burton, Donald E.},

abstractNote = {In this paper, we present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem in 2D Cartesian geometry. This area-weighted approach, in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. Finally, a suite of test problems are calculated to demonstrate stable mesh motion, the expected second order accuracy of these methods, and that the new area-weighted DG method preserves spherical symmetry on 1D radial flow problems with equal-angle polar meshes.},

doi = {10.1016/j.jcp.2018.06.073},

journal = {Journal of Computational Physics},

number = ,

volume = 373,

place = {United States},

year = {2018},

month = {7}

}

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