# Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Balance Laws and Their Weak Local Residuals

## 3. Shallow Water Equations and Their Weak Local Residuals

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 4. Numerical Tests

#### 4.1. A One-Dimensional Problem with Topography

#### 4.2. A One-Dimensional Dam Break with Passive Tracer

#### 4.3. A Planar Dam Break in Two Dimensions

#### 4.4. A Radial Dam Break in Two Dimensions

**Remark**

**4.**

**Remark**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1D | one dimension |

1.5D | one and a half dimensions |

2D | two dimensions |

CFL | Courant–Friedrichs–Lewy |

CK | Constantin–Kurganov |

iFEM | integrated finite element method |

KNP | Kurganov–Noelle–Petrova |

SI | Systéme International (International System) |

SWE | shallow water equations |

WLR | weak local residual |

## References

- Lee, S. Dynamics of trapped solitary waves for the forced KdV equation. Symmetry
**2018**, 10, 129. [Google Scholar] [CrossRef] [Green Version] - Shi, D.; Zhang, Y.; Liu, W.; Liu, J. Some exact solutions and conservation laws of the coupled time-fractional Boussinesq-Burgers system. Symmetry
**2019**, 11, 77. [Google Scholar] [CrossRef] [Green Version] - Caputo, J.-G.; Dutykh, D.; Gleyse, B. Coupling conditions for water waves at forks. Symmetry
**2019**, 11, 434. [Google Scholar] [CrossRef] [Green Version] - Paliathanasis, A. One-dimensional optimal system for 2D rotating ideal gas. Symmetry
**2019**, 11, 1115. [Google Scholar] [CrossRef] [Green Version] - Castaings, W.; Navon, I.M. Mesh refinement strategies for solving singularly perturbed reaction-diffusion problems. Comput. Math. Appl.
**2001**, 41, 157–176. [Google Scholar] [CrossRef] [Green Version] - Anco, S.C. On the incompleteness of Ibragimov’s conservation law theorem and its equivalence to a standard formula using symmetries and adjoint-symmetries. Symmetry
**2017**, 9, 33. [Google Scholar] [CrossRef] [Green Version] - Bravo, J.C.; Castilla, M.V. Energy conservation law in industrial architecture: An approach through geometric algebra. Symmetry
**2016**, 8, 92. [Google Scholar] [CrossRef] - Buhe, E.; Bluman, G.W.; Alatancang, C.; Yulan, H. Some approaches to the calculation of conservation laws for a telegraph system and their comparisons. Symmetry
**2018**, 10, 182. [Google Scholar] [CrossRef] [Green Version] - Recio, E.; Garrido, T.M.; de la Rosa, R.; Bruzón, M.S. Hamiltonian structure, symmetries and conservation laws for a generalized (2 + 1)-dimensional double dispersion equation. Symmetry
**2019**, 11, 1031. [Google Scholar] [CrossRef] [Green Version] - Ebrahimpour, M.; Shafaghat, R.; Alamian, R.; Shadloo, M.S. Numerical investigation of the Savonius vertical axis wind turbine and evaluation of the effect of the overlap parameter in both horizontal and vertical directions on its performance. Symmetry
**2019**, 11, 821. [Google Scholar] [CrossRef] [Green Version] - Jalali, E.; Akbari, O.A.; Sarafraz, M.M.; Abbas, T.; Safaei, M.R. Heat transfer of oil/MWCNT nanofluid jet injection inside a rectangular microchannel. Symmetry
**2019**, 11, 757. [Google Scholar] [CrossRef] [Green Version] - Masuda, T.; Tagawa, T. Quasi-periodic oscillating flows in a channel with a suddenly expanded section. Symmetry
**2019**, 11, 1403. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; Gerdt, V.P.; Blinkov, Y.A. Algebraic construction of a strongly consistent, permutationally symmetric and conservative difference scheme for 3D steady Stokes flow. Symmetry
**2019**, 11, 269. [Google Scholar] [CrossRef] [Green Version] - Karni, S.; Kurganov, A. Local error analysis for approximate solutions of hyperbolic conservation laws. Adv. Comput. Math.
**2005**, 22, 79–99. [Google Scholar] [CrossRef] - Karni, S.; Kurganov, A.; Petrova, G. A smoothness indicator for adaptive algorithms for hyperbolic systems. J. Comput. Phys.
**2002**, 178, 323–341. [Google Scholar] [CrossRef] [Green Version] - Constantin, L.A.; Kurganov, A. Adaptive central-upwind schemes for hyperbolic systems of conservation laws. In Hyperbolic Problems: Theory, Numerics, and Applications (Osaka, Japan, 2004); Asakura, F., Kawashima, S., Matsumura, A., Nishibata, S., Nishihara, K., Eds.; Yokohama Publishers: Yokohama, Japan, 2006; Volume 1, pp. 95–103. [Google Scholar]
- Puppo, G.; Semplice, M. Numerical entropy and adaptivity for finite volume schemes. Commun. Comput. Phys.
**2011**, 10, 1132–1160. [Google Scholar] [CrossRef] [Green Version] - Puppo, G. Numerical entropy production on shocks and smooth transitions. J. Sci. Comput.
**2002**, 17, 263–271. [Google Scholar] [CrossRef] - Puppo, G. Numerical entropy production for central schemes. SIAM J. Sci. Comput.
**2003**, 25, 1382–1415. [Google Scholar] [CrossRef] - Mungkasi, S. A Study of Well-Balanced Finite Volume Methods and Refinement Indicators for the Shallow Water Equations. Ph.D. Thesis, The Australian National University, Canberra, ACT, Australia, 2012. [Google Scholar]
- Mungkasi, S. Adaptive finite volume method for the shallow water equations on triangular grids. Adv. Math. Phys.
**2016**, 2016, 7528625. [Google Scholar] [CrossRef] - Felcman, J.; Kadrnka, L. Adaptive finite volume approximation of the shallow water equations. Appl. Math. Comput.
**2012**, 219, 3354–3366. [Google Scholar] [CrossRef] - Mungkasi, S.; Roberts, S.G. Weak local residuals in an adaptive finite volume method for one-dimensional shallow water equations. J. Indones. Math. Soc.
**2014**, 20, 11–18. [Google Scholar] [CrossRef] - Mungkasi, S.; Roberts, S.G. Well-balanced computations of weak local residuals of the shallow water equations. ANZIAM J.
**2015**, 56, C128–C147. [Google Scholar] [CrossRef] [Green Version] - Mungkasi, S.; Li, Z.; Roberts, S.G. Weak local residuals as smoothness indicators for the shallow water equations. Appl. Math. Lett.
**2014**, 30, 51–55. [Google Scholar] [CrossRef] [Green Version] - Knobel, R. An Introduction to the Mathematical Theory of Waves; American Mathematical Society: Providence, RI, USA, 2000. [Google Scholar]
- Smoller, J. Shock Waves and Reaction-Diffusion Equations; Springer: New York, NY, USA, 1983. [Google Scholar]
- Mungkasi, S.; Roberts, S.G. Numerical entropy production for shallow water flows. ANZIAM J.
**2011**, 52, C1–C17. [Google Scholar] [CrossRef] [Green Version] - Chen, L. iFEM: An Integrated Finite Element Method Package in MATLAB; Technical Report; University of California: Irvine, CA, USA, 2009. [Google Scholar]
- Chen, L.; Zhang, C.-S. A coarsening algorithm on adaptive grids by newest vertex bisection and its applications. J. Comput. Math.
**2010**, 28, 767–789. [Google Scholar] [CrossRef] - Kurganov, A.; Noelle, S.; Petrova, G. Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comput.
**2001**, 23, 707–740. [Google Scholar] [CrossRef] [Green Version] - Mungkasi, S.; Roberts, S.G. On the best quantity reconstructions for a well balanced finite volume method used to solve the shallow water wave equations with a wet/dry interface. ANZIAM J.
**2010**, 51, C48–C65. [Google Scholar] [CrossRef] [Green Version] - Audusse, E.; Bouchut, F.; Bristeau, M.O.; Klein, R.; Perthame, B. A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput.
**2004**, 25, 2050–2065. [Google Scholar] [CrossRef] - Noelle, S.; Pankratz, N.; Puppo, G.; Natvig, J.R. Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys.
**2006**, 213, 474–499. [Google Scholar] [CrossRef] - Bouchut, F. Efficient numerical finite volume schemes for shallow water models. In Nonlinear Dynamics of Rotating Shallow Water: Methods and Advances; Edited Series on Advances in Nonlinear Science and Complexity; Zeitlin, V., Ed.; Elsevier: Amsterdam, The Netherlands, 2007; Volume 2, pp. 189–256. [Google Scholar]
- Harten, A. The artificial compression method for computation of shocks and contact discontinuities I: Single conservation laws. Commun. Pure Appl. Math.
**1977**, 30, 611–638. [Google Scholar] [CrossRef]

**Figure 1.**Flow over a parabolic bump at time $t=100$ using the adaptive method. For this figure, we enforce that the maximum level of refinement is 5.

**Figure 2.**Flow with passive tracer at time $t=90$ after dam break. CK is defined as in Equation (17). ‘Ana sol’ means the analytical solution. ‘Num sol’ is the numerical solution. Both solutions coincide.

**Figure 3.**Example of refining the mesh twice around the northeast corner node ${N}_{3}(1,1)$ of the domain.

**Figure 4.**The left subfigure is the water surface and the right one is $\left|\mathrm{CK}\right|/\mathrm{max}\left\{\left|\mathrm{CK}\right|\right\}$ at time $t=0.2$ after the planar dam is broken.

**Figure 5.**The left subfigure is the x-momentum and the right one is the corresponding mesh at time $t=0.2$ after the planar dam is broken.

**Figure 6.**The left subfigure is the water surface and the right one is $\left|\mathrm{CK}\right|/\mathrm{max}\left\{\left|\mathrm{CK}\right|\right\}$ at time $t=0.05$ after the radial dam is broken.

**Figure 7.**The left subfigure is the x-momentum and the right one is the corresponding mesh at time $t=0.05$ after the radial dam is broken.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mungkasi, S.; Roberts, S.G.
Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows. *Symmetry* **2020**, *12*, 345.
https://doi.org/10.3390/sym12030345

**AMA Style**

Mungkasi S, Roberts SG.
Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows. *Symmetry*. 2020; 12(3):345.
https://doi.org/10.3390/sym12030345

**Chicago/Turabian Style**

Mungkasi, Sudi, and Stephen Gwyn Roberts.
2020. "Weak Local Residuals as Smoothness Indicators in Adaptive Mesh Methods for Shallow Water Flows" *Symmetry* 12, no. 3: 345.
https://doi.org/10.3390/sym12030345