### Abstract

Iterative solution procedures for solving the complete mild-slope wave (combined refraction-diffraction) equation are developed. Existing models for investigating wave refraction-diffraction in coastal areas have one of two main problems; (i) Some of the physics is lost as they resort to approximate solutions (e.g. parabolic approximations). Thus they are inappropriate in many situations. (ii) If all of the physics is to be incorporated, the problem defies computer solution except for extremely small domains (approximately 10 wavelengths), chiefly because the matrix equation associated with numerical discretization of the complete problem does not normally lend itself to solution by iteration. This paper describes the construction of iterative models that overcome both problems. First, a modified equation with an identical solution but which lends itself to iterative procedures is formulated, and the conjugate gradient method is used. A second, more rapidly converging algorithm is obtained by preconditioning. It is shown that the algorithms can be conveniently implemented on regions much larger thanthose handled by conventional models, without compromising the physics of the equation. Further, they can be efficiently run in either the linear or nonlinear mode. Comparisons of model results with laboratory data and other numerical and analytical solutions are found to be excellent for several cases. The procedures thus enable the engineer to expand the scope of the mild-slope equation. As an example, an experiment is performed to demonstrate the sensitivity of the wavefield to the location of a breakwater in a region with complex bathymetry.

Original language | English |
---|---|

Pages (from-to) | 187-199 |

Number of pages | 13 |

Journal | Applied Ocean Research |

Volume | 13 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1991 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Ocean Engineering

### Cite this

*Applied Ocean Research*,

*13*(4), 187-199. https://doi.org/10.1016/S0141-1187(05)80074-4

**Solution of the mild-slope wave problem by iteration.** / Panchang, Vijay; Pearce, Bryan R.; Wei, Ge; Cushman-Roisin, Benoit.

Research output: Contribution to journal › Article

*Applied Ocean Research*, vol. 13, no. 4, pp. 187-199. https://doi.org/10.1016/S0141-1187(05)80074-4

}

TY - JOUR

T1 - Solution of the mild-slope wave problem by iteration

AU - Panchang, Vijay

AU - Pearce, Bryan R.

AU - Wei, Ge

AU - Cushman-Roisin, Benoit

PY - 1991

Y1 - 1991

N2 - Iterative solution procedures for solving the complete mild-slope wave (combined refraction-diffraction) equation are developed. Existing models for investigating wave refraction-diffraction in coastal areas have one of two main problems; (i) Some of the physics is lost as they resort to approximate solutions (e.g. parabolic approximations). Thus they are inappropriate in many situations. (ii) If all of the physics is to be incorporated, the problem defies computer solution except for extremely small domains (approximately 10 wavelengths), chiefly because the matrix equation associated with numerical discretization of the complete problem does not normally lend itself to solution by iteration. This paper describes the construction of iterative models that overcome both problems. First, a modified equation with an identical solution but which lends itself to iterative procedures is formulated, and the conjugate gradient method is used. A second, more rapidly converging algorithm is obtained by preconditioning. It is shown that the algorithms can be conveniently implemented on regions much larger thanthose handled by conventional models, without compromising the physics of the equation. Further, they can be efficiently run in either the linear or nonlinear mode. Comparisons of model results with laboratory data and other numerical and analytical solutions are found to be excellent for several cases. The procedures thus enable the engineer to expand the scope of the mild-slope equation. As an example, an experiment is performed to demonstrate the sensitivity of the wavefield to the location of a breakwater in a region with complex bathymetry.

AB - Iterative solution procedures for solving the complete mild-slope wave (combined refraction-diffraction) equation are developed. Existing models for investigating wave refraction-diffraction in coastal areas have one of two main problems; (i) Some of the physics is lost as they resort to approximate solutions (e.g. parabolic approximations). Thus they are inappropriate in many situations. (ii) If all of the physics is to be incorporated, the problem defies computer solution except for extremely small domains (approximately 10 wavelengths), chiefly because the matrix equation associated with numerical discretization of the complete problem does not normally lend itself to solution by iteration. This paper describes the construction of iterative models that overcome both problems. First, a modified equation with an identical solution but which lends itself to iterative procedures is formulated, and the conjugate gradient method is used. A second, more rapidly converging algorithm is obtained by preconditioning. It is shown that the algorithms can be conveniently implemented on regions much larger thanthose handled by conventional models, without compromising the physics of the equation. Further, they can be efficiently run in either the linear or nonlinear mode. Comparisons of model results with laboratory data and other numerical and analytical solutions are found to be excellent for several cases. The procedures thus enable the engineer to expand the scope of the mild-slope equation. As an example, an experiment is performed to demonstrate the sensitivity of the wavefield to the location of a breakwater in a region with complex bathymetry.

UR - http://www.scopus.com/inward/record.url?scp=0026097835&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026097835&partnerID=8YFLogxK

U2 - 10.1016/S0141-1187(05)80074-4

DO - 10.1016/S0141-1187(05)80074-4

M3 - Article

AN - SCOPUS:0026097835

VL - 13

SP - 187

EP - 199

JO - Applied Ocean Research

JF - Applied Ocean Research

SN - 0141-1187

IS - 4

ER -