# Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes

@article{Xing2013PositivityPreservingWD, title={Positivity-Preserving Well-Balanced Discontinuous Galerkin Methods for the Shallow Water Equations on Unstructured Triangular Meshes}, author={Yulong Xing and Xiangxiong Zhang}, journal={Journal of Scientific Computing}, year={2013}, volume={57}, pages={19-41} }

The shallow water equations model flows in rivers and coastal areas and have wide applications in ocean, hydraulic engineering, and atmospheric modeling. In “Xing et al. Adv. Water Resourc. 33: 1476–1493, 2010)”, the authors constructed high order discontinuous Galerkin methods for the shallow water equations which can maintain the still water steady state exactly, and at the same time can preserve the non-negativity of the water height without loss of mass conservation. In this paper, we…

## 87 Citations

Well-Balanced Discontinuous Galerkin Method for Shallow Water Equations with Constant Subtraction Techniques on Unstructured Meshes

- Computer ScienceJ. Sci. Comput.
- 2019

A high-order accurate and well-balanced discontinuous Galerkin (DG) method on two dimensional (2D) unstructured meshes for the Saint–Venant shallow water equations and Hierarchical reconstruction limiter with a remainder correction technique is introduced to control numerical oscillations.

Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

- Computer ScienceJ. Sci. Comput.
- 2020

This work proposes entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography, and establishes an entropy stable scheme by adding additional dissipative terms.

Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms

- Computer Science
- 2014

Application of positivity-preserving well-balanced discontinuous Galerkin method in computational hydrology

- Computer Science
- 2016

Entropy Stable andWell-Balanced Discontinuous Galerkin Methods for the Nonlinear ShallowWater Equations

- Computer Science
- 2020

This work proposes entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography, and establishes an entropy stable scheme by adding additional dissipative terms.

A well-balanced moving mesh discontinuous Galerkin method for the Ripa model on triangular meshes

- Environmental ScienceSSRN Electronic Journal
- 2022

A well-balanced moving mesh discontinuous Galerkin (DG) method is proposed for the numerical solution of the Ripa model – a generalization of the shallow water equations that accounts for eﬀects of…

Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction

- Computer ScienceJ. Comput. Phys.
- 2018

A high-order well-balanced positivity-preserving moving mesh DG method for the shallow water equations with non-flat bottom topography

- Environmental ScienceJ. Sci. Comput.
- 2021

A rezoning-type adaptive moving mesh discontinuous Galerkin method is proposed for the numerical solution of the shallow water equations with non-flat bottom topography and its ability to capture small perturbations of the lake-at-rest steady state is demonstrated.

A Survey of High Order Schemes for the Shallow Water Equations

- Mathematics
- 2014

In this paper, we survey our recent work on designing high order positivity- preserving well-balanced finite difference and finite volume WENO (weighted essen- tially non-oscillatory) schemes, and…

High order well-balanced discontinuous Galerkin methods based on hydrostatic reconstruction for shallow water equations

- Computer ScienceJ. Comput. Appl. Math.
- 2018

## References

SHOWING 1-10 OF 48 REFERENCES

Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations

- Computer Science
- 2010

A Discontinuous Galerkin Global Shallow Water Model

- Computer Science
- 2005

A discontinuous Galerkin shallow water model on the cubed sphere is developed, thereby extending the transport scheme developed by Nair et al. and conservation of these quantities is better preserved than in existing finite-volume models.

High-order finite volume WENO schemes for the shallow water equations with dry states

- Environmental Science
- 2011

Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations

- Mathematics
- 2002

We present a high-order discontinuous Galerkin method for the solution of the shallow water equations on the sphere. To overcome well-known problems with polar singularities, we consider the shallow…

A triangular spectral/hp discontinuous Galerkin method for modelling 2D shallow water equations

- Mathematics
- 2004

We present a spectral/hp element discontinuous Galerkin model for simulating shallow water flows on unstructured triangular meshes. The model uses an orthogonal modal expansion basis of arbitrary…

Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

- Computer Science
- 2011

A new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term and a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows

- MathematicsSIAM J. Sci. Comput.
- 2004

A general strategy is described, based on a local hydrostatic reconstruction, that allows a well-balanced scheme to derive from any given numerical flux for the homogeneous problem, whenever the initial solver satisfies some classical stability properties.

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

- Mathematics
- 2008

We build and analyze a Runge–Kutta discontinuous Galerkin method to approximate the one‐ and two‐dimensional shallow‐water equations. We introduce a flux modification technique to derive a…