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## A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme

## Abstract

We present here a Lagrangian discontinuous Galerkin (DG) hydrodynamic method that is up to third-order accurate using subcell mesh stabilization (SMS) for compressible flows on quadratic meshes in two-dimensional (2D) Cartesian coordinates. Similar to the second-order accurate Lagrangian DG method with linear meshes, the physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with Taylor series polynomials. The Riemann velocity at the vertices of a curvilinear cell, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem. Curvilinear cells (e.g., quadratic quadrilateral meshes in this work) have many deformational degrees of freedom, and with these cells, they can deform in unphysical ways. Likewise, the Riemann solution at an edge vertex differs from the one at the corner of a cell. With SMS, each quadratic quadrilateral cell is decomposed into four quadrilateral subcells, that move in a Lagrangian manner. The edge vertex is surrounded by four subcells so that it is similar to the vertex at the cell corner. SMS can detect inconsistent density fields between the cell and subcells. The difference between these two density fields is used to correct the stress (pressure) input to themore »

- 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)

- OSTI Identifier:
- 1542835

- Report Number(s):
- LA-UR-18-26689

Journal ID: ISSN 0021-9991

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 386; Journal ID: ISSN 0021-9991

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 96 KNOWLEDGE MANAGEMENT AND PRESERVATION; Lagrangian shock hydrodynamics; discontinuous Galerkin; cell-centered hydrodynamics; third-order accurate; quadratic curvilinear cells

### Citation Formats

```
Liu, Xiaodong, Morgan, Nathaniel R., and Burton, Donald E. A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.02.008.
```

```
Liu, Xiaodong, Morgan, Nathaniel R., & Burton, Donald E. A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme. United States. doi:10.1016/j.jcp.2019.02.008.
```

```
Liu, Xiaodong, Morgan, Nathaniel R., and Burton, Donald E. Wed .
"A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme". United States. doi:10.1016/j.jcp.2019.02.008.
```

```
@article{osti_1542835,
```

title = {A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme},

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

abstractNote = {We present here a Lagrangian discontinuous Galerkin (DG) hydrodynamic method that is up to third-order accurate using subcell mesh stabilization (SMS) for compressible flows on quadratic meshes in two-dimensional (2D) Cartesian coordinates. Similar to the second-order accurate Lagrangian DG method with linear meshes, the physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with Taylor series polynomials. The Riemann velocity at the vertices of a curvilinear cell, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem. Curvilinear cells (e.g., quadratic quadrilateral meshes in this work) have many deformational degrees of freedom, and with these cells, they can deform in unphysical ways. Likewise, the Riemann solution at an edge vertex differs from the one at the corner of a cell. With SMS, each quadratic quadrilateral cell is decomposed into four quadrilateral subcells, that move in a Lagrangian manner. The edge vertex is surrounded by four subcells so that it is similar to the vertex at the cell corner. SMS can detect inconsistent density fields between the cell and subcells. The difference between these two density fields is used to correct the stress (pressure) input to the Riemann solver. This SMS scheme enables stable mesh motion and accurate solutions in the context of a Lagrangian high-order DG method that is up to third-order with quadratic cells. We also present effective limiting strategies that ensure monotonicity of the primitive variables with the high-order DG method. This Lagrangian DG hydrodynamic method with SMS conserves mass, momentum, and total energy. A suite of test problems are calculated to demonstrate the designed order of accuracy (up to third-order accurate) of this method, and that the Lagrangian DG method using SMS preserves cylindrical symmetry on 1D radial flow problems with an equal-angle polar quadratic mesh.},

doi = {10.1016/j.jcp.2019.02.008},

journal = {Journal of Computational Physics},

number = ,

volume = 386,

place = {United States},

year = {2019},

month = {2}

}

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