# A Lagrangian discontinuous Galerkin hydrodynamic method

## Abstract

Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The third approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.

- 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 Laboratory Directed Research and Development (LDRD) Program

- OSTI Identifier:
- 1414156

- Report Number(s):
- LA-UR-17-24361

Journal ID: ISSN 0045-7930

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computers and Fluids

- Additional Journal Information:
- Journal Volume: 163; Journal ID: ISSN 0045-7930

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Lagrangian; Hydrodynamics; Discontinuous Galerkin; Taylor basis; cell-centered; compressible flows; shocks

### Citation Formats

```
Liu, Xiaodong, Morgan, Nathaniel Ray, and Burton, Donald E.
```*A Lagrangian discontinuous Galerkin hydrodynamic method*. United States: N. p., 2017.
Web. doi:10.1016/j.compfluid.2017.12.007.

```
Liu, Xiaodong, Morgan, Nathaniel Ray, & Burton, Donald E.
```*A Lagrangian discontinuous Galerkin hydrodynamic method*. United States. doi:10.1016/j.compfluid.2017.12.007.

```
Liu, Xiaodong, Morgan, Nathaniel Ray, and Burton, Donald E. Mon .
"A Lagrangian discontinuous Galerkin hydrodynamic method". United States.
doi:10.1016/j.compfluid.2017.12.007.
```

```
@article{osti_1414156,
```

title = {A Lagrangian discontinuous Galerkin hydrodynamic method},

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

abstractNote = {Here, we present a new Lagrangian discontinuous Galerkin (DG) hydrodynamic method for solving the two-dimensional gas dynamic equations on unstructured hybrid meshes. The physical conservation laws for the momentum and total energy are discretized using a DG method based on linear Taylor expansions. Three different approaches are investigated for calculating the density variation over the element. The first approach evolves a Taylor expansion of the specific volume field. The second approach follows certain finite element methods and uses the strong mass conservation to calculate the density field at a location inside the element or on the element surface. The third approach evolves a Taylor expansion of the density field. The nodal velocity, and the corresponding forces, are explicitly calculated by solving a multidirectional approximate Riemann problem. An effective limiting strategy is presented that ensures monotonicity of the primitive variables. This new Lagrangian DG hydrodynamic method conserves mass, momentum, and total energy. Results from a suite of test problems are presented to demonstrate the robustness and expected second-order accuracy of this new method.},

doi = {10.1016/j.compfluid.2017.12.007},

journal = {Computers and Fluids},

number = ,

volume = 163,

place = {United States},

year = {Mon Dec 11 00:00:00 EST 2017},

month = {Mon Dec 11 00:00:00 EST 2017}

}

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