Method of lines solutions of the extended Boussinesq equations.

*(English)*Zbl 1077.65104Summary: A numerical solution procedure based on the method of lines for solving the alternative form of the one-dimensional extended Boussinesq equations studied by O. Nwogu [Alternative form of Boussinesq equations for nearshove wave propagation. ASCE J. Waterw. Port. Coast. Ocean Eng. 119, 618–638 (1993)] is presented. The numerical scheme is accurate up to fifth-order in time and fourth-order accurate in space, thus reducing all truncation errors to a level smaller than the dispersive terms retained by most extended Boussinesq models. Exact solitary wave solutions and invariants of motions recently derived by the authors are used to specify initial data for the incident solitary waves in the numerical model of Nwogu and for the verification of the associated computed solutions. The invariants of motions and several error measures are monitored in order to assess the conservative properties and the accuracy of the numerical scheme. The proposed method of lines solution procedure is general and can be easily modified to solve a wide range of Boussinesq-like equations in coastal engineering.

##### MSC:

65M20 | Method of lines for initial value and initial-boundary value problems involving PDEs |

35K55 | Nonlinear parabolic equations |

35Q35 | PDEs in connection with fluid mechanics |

##### Keywords:

Method of lines; Boussinesq equations; Nonlinearity; Dispersion; Solitary wave solutions; Invariants of motion; numerical examples; coastal engineering
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\textit{S. Hamdi} et al., J. Comput. Appl. Math. 183, No. 2, 327--342 (2005; Zbl 1077.65104)

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