Modelling an undular bore with \(B\)-splines. (English) Zbl 0897.76049
Summary: The development of an undular bore is governed by the regularized long wave equation. For its solution, we present a Petrov-Galerkin B-spline finite element method and validate the numerical model by considering the motion of a single solitary wave. The development of an undular bore is studied through computer simulation.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76D33 | Waves for incompressible viscous fluids |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
regularized long wave equation; Petrov-Galerkin B-spline finite element method; solitary waveReferences:
| [1] | Peregrine, D. H., Calculations of the development of an undular bore, J. Fluid Mech., 25, 321-330 (1966) |
| [2] | Benjamin, T. B.; Bona, J. L.; Mohony, J. J., Model equations for long waves in non-linear dispersive systems, Phil. Trans. Roy. Soc., 272A, 47-78 (1972) · Zbl 0229.35013 |
| [3] | Olver, P. J., Euler operators and conservation laws of the BBM equation, (Math. Proc. Camb. Phil. Soc., 85 (1979)), 143-159 · Zbl 0387.35050 |
| [4] | Gardner, G. A.; Gardner, L. R.T.; Ali, A. H.A., Modelling non-linear waves with B-spline finite elements, (Cohen, G.; Halpern, L.; Joly, P., Math, and Num. Aspects of Wave Propagation (1991), SIAM: SIAM Philadelphia), 533-542 · Zbl 0883.65075 |
| [5] | Prenter, P. M., Splines and Variation Methods (1975), J. Wiley · Zbl 0344.65044 |
| [6] | Gardner, L. R.T.; Gardner, G. A., Solitary waves of the RLW equation, J. Comput. Phys., 91, 441-459 (1990) · Zbl 0717.65072 |
| [7] | Jain, P. C.; Shankar, R.; Singh, T. V., Numerical solution of the RLW equation, Comm. Numer. Methods Engrg., 9, 587-594 (1993) · Zbl 0777.76049 |
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