# A high-order discontinuous Galerkin method for 2D incompressible flows

## Abstract

In this paper the authors introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total entropy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Maryland, College Park, MD (US)

- OSTI Identifier:
- 20067699

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 160; Journal Issue: 2; Other Information: PBD: 20 May 2000; Journal ID: ISSN 0021-9991

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; INCOMPRESSIBLE FLOW; GALERKIN-PETROV METHOD; TWO-DIMENSIONAL CALCULATIONS; IDEAL FLOW; TIME DEPENDENCE

### Citation Formats

```
Liu, J.G., and Shu, C.W.
```*A high-order discontinuous Galerkin method for 2D incompressible flows*. United States: N. p., 2000.
Web. doi:10.1006/jcph.2000.6475.

```
Liu, J.G., & Shu, C.W.
```*A high-order discontinuous Galerkin method for 2D incompressible flows*. United States. doi:10.1006/jcph.2000.6475.

```
Liu, J.G., and Shu, C.W. Sat .
"A high-order discontinuous Galerkin method for 2D incompressible flows". United States. doi:10.1006/jcph.2000.6475.
```

```
@article{osti_20067699,
```

title = {A high-order discontinuous Galerkin method for 2D incompressible flows},

author = {Liu, J.G. and Shu, C.W.},

abstractNote = {In this paper the authors introduce a high-order discontinuous Galerkin method for two-dimensional incompressible flow in the vorticity stream-function formulation. The momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The stream function is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total entropy stability. The method is efficient for inviscid or high Reynolds number flows. Optimal error estimates are proved and verified by numerical experiments.},

doi = {10.1006/jcph.2000.6475},

journal = {Journal of Computational Physics},

issn = {0021-9991},

number = 2,

volume = 160,

place = {United States},

year = {2000},

month = {5}

}