A robust, finite element model for hydrostatic surface water flows.

*(English)*Zbl 0915.76056Summary: We introduce a finite element scheme for the two-dimensional shallow water equations using semi-implicit methods in time. A semi-Lagrangian method is used to approximate the effects of advection. A wave equation is formed at the discrete level, such that the equations decouple into an equation for surface elevation and a momentum equation for the horizontal velocity. We examine the convergence rates and relative computational efficiency with the use of three test cases representing various degrees of difficulty. A test with a polar-quadrant grid investigates the response to local grid-scale forcing and the presence of spurious modes, a channel test case establishes convergence rates, and a field-scale test case examines problems with highly irregular grids.

##### MSC:

76M10 | Finite element methods applied to problems in fluid mechanics |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

86A05 | Hydrology, hydrography, oceanography |

##### Keywords:

two-dimensional shallow water equations; semi-implicit methods in time; semi-Lagrangian method; effects of advection; wave equation; convergence rates; polar-quadrant grid; local grid-scale forcing; spurious modes; channel test; field-scale test
PDF
BibTeX
XML
Cite

\textit{R. A. Walters} and \textit{V. Casulli}, Commun. Numer. Methods Eng. 14, No. 10, 931--940 (1998; Zbl 0915.76056)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Gray, Time-stepping schemes for finite element tidal model computations, Adv. Water Resour. 1 pp 83– (1977) |

[2] | Gray, Some inadequacies of finite element models as simulators for two-dimensional circulation, Adv. Water Resour. 5 pp 171– (1982) |

[3] | Platzman, Some response characteristics of finite element tidal models, J. Comput. Phys. 40 pp 36– (1981) · Zbl 0469.76016 |

[4] | Walters, Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations, Comput. Fluids 11 pp 51– (1983) · Zbl 0521.76017 |

[5] | Walters, Numerically induced oscillations in finite element approximations to the shallow water equations, Int. j. numer. methods fluids 3 pp 591– (1983) · Zbl 0526.76028 |

[6] | Fortin, Old and new finite elements for incompressible flows, Int. J. Numer. Methods Fluids 1 pp 347– (1981) · Zbl 0467.76030 |

[7] | Lynch, A wave equation model for finite element tidal computations, Comput. Fluids 7 pp 207– (1979) · Zbl 0421.76013 |

[8] | Lynch, Three dimensional hydrodynamics on finite elements. Part 2: Nonlinear time stepping model, Int. j. numer. methods fluids 12 pp 507– (1991) · Zbl 0722.76037 |

[9] | Bova, Finite Element Modeling of Environmental Problems pp 115– (1995) |

[10] | Kolar, Aspects of nonlinear simulation using shallow-water models based on the wave continuity equation, Comput. Fluids 23 pp 523– (1994) · Zbl 0813.76042 |

[11] | Walters, A three-dimensional, finite element model for coastal and estuarine circulation, Continent. Shelf Res. 12 pp 83– (1992) |

[12] | Foreman, A tidal model for eastern Juan de Fuca Strait and the southern Strait of Georgia, J. Geophys. Res. 100 pp 721– (1995) |

[13] | Pinder, Finite Elements in Surface and Substarface Hydrology (1977) |

[14] | Becker, Finite Elements: An Introduction (1981) |

[15] | Arbogast, Finite Element Modeling of Environmental Problems pp 275– (1995) |

[16] | Casulli, Semi-implicit finite difference methods for the two-dimensional shallow water equations, J. Comput. Phys. 86 pp 56– (1990) · Zbl 0681.76022 |

[17] | Staniforth, Semi-Lagrangian integration schemes for atmospheric models - A review, Mon. Weather Rev. 119 pp 2206– (1991) |

[18] | Casulli, Stability, accuracy, and efficiency of a semi-implicit method for three-dimensional shallow water flow, Comput. Math. Appl. 27 pp 99– (1994) · Zbl 0796.76052 |

[19] | Walters, Accuracy of an estuarine hydrodynamic model using smooth elements, Water Resour. Res. 16 pp 187– (1980) |

[20] | Walters, Comparison of h and p finite element approximations of the shallow water equations, Int. j. numer. methods fluids 16 (1996) |

[21] | Henry, A geometrically-based, automatic generator for irregular triangular networks, Commun. numer. methods eng. 9 pp 555– (1993) · Zbl 0782.65139 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.