# An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry

## Abstract

We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.

- Authors:

- Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln (Germany)
- Department of Mathematics, The Florida State University, Tallahassee, FL 32306 (United States)

- Publication Date:

- OSTI Identifier:
- 22622301

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 340; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; APPROXIMATIONS; BALANCES; BATHYMETRY; ENTROPY; EQUATIONS; INTERFACES; MASS; NONLINEAR PROBLEMS; PYRAZOLINES; SOURCE TERMS; STABILITY; THERMODYNAMICS; TWO-DIMENSIONAL CALCULATIONS; TWO-DIMENSIONAL SYSTEMS; WATER

### Citation Formats

```
Wintermeyer, Niklas, Winters, Andrew R., E-mail: awinters@math.uni-koeln.de, Gassner, Gregor J., and Kopriva, David A.
```*An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.03.036.

```
Wintermeyer, Niklas, Winters, Andrew R., E-mail: awinters@math.uni-koeln.de, Gassner, Gregor J., & Kopriva, David A.
```*An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry*. United States. doi:10.1016/J.JCP.2017.03.036.

```
Wintermeyer, Niklas, Winters, Andrew R., E-mail: awinters@math.uni-koeln.de, Gassner, Gregor J., and Kopriva, David A. Sat .
"An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry". United States.
doi:10.1016/J.JCP.2017.03.036.
```

```
@article{osti_22622301,
```

title = {An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry},

author = {Wintermeyer, Niklas and Winters, Andrew R., E-mail: awinters@math.uni-koeln.de and Gassner, Gregor J. and Kopriva, David A.},

abstractNote = {We design an arbitrary high-order accurate nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations with non-constant, possibly discontinuous, bathymetry on unstructured, possibly curved, quadrilateral meshes. The scheme is derived from an equivalent flux differencing formulation of the split form of the equations. We prove that this discretization exactly preserves the local mass and momentum. Furthermore, combined with a special numerical interface flux function, the method exactly preserves the mathematical entropy, which is the total energy for the shallow water equations. By adding a specific form of interface dissipation to the baseline entropy conserving scheme we create a provably entropy stable scheme. That is, the numerical scheme discretely satisfies the second law of thermodynamics. Finally, with a particular discretization of the bathymetry source term we prove that the numerical approximation is well-balanced. We provide numerical examples that verify the theoretical findings and furthermore provide an application of the scheme for a partial break of a curved dam test problem.},

doi = {10.1016/J.JCP.2017.03.036},

journal = {Journal of Computational Physics},

number = ,

volume = 340,

place = {United States},

year = {Sat Jul 01 00:00:00 EDT 2017},

month = {Sat Jul 01 00:00:00 EDT 2017}

}