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Article information
2015 , Volume 20, ¹ 4, p.83-106
Shokina N.Y., Khakimzyanov G.S.
An improved adaptive grid method for one-dimensional shallow water equations
An improved adaptive grid method is considered for the numerical solution of the problems on propagation and run-up of surface waves, described by the one-dimensional shallow water model. The modified algorithm for the realization of the explicit predictorcorrector scheme is presented, which is based on the new way of computation of the right-hand side of the shallow water equations. The algorithm provides savings in computational time in comparison with its earlier version while preserving the approximation order. Also the preservation of the state of rest is guaranteed in transition from one time level to a next one. A new method for choosing the scheme parameters on the basis of the analysis of the differential approximation is suggested that guarantees the satisfaction of the TVD-property for the improved predictor-corrector scheme. The presented method for construction of different conservative schemes on moving grids is based on an appropriate choice of the scheme parameters for the predictor-corrector scheme, which represents the canonical form of the two-layer explicit schemes for the shallow water equations. As an example, a conservative upwind scheme on moving grid is provided in the divergent and non-divergent forms. The properties of the upwind scheme and the predictor-corrector scheme on dynamically adaptive grids are demonstrated for the exact solution of the nonlinear shallow water equations. Using the known analytical solutions of the shallow water equations in the vicinity of the water-land boundary the improved difference boundary conditions are obtained at the moving waterfront point. These boundary conditions approximate the analytical solutions with a higher accuracy than the conditions used in the earlier works. It is proved that if a fluid is at rest and has a non-perturbed free boundary at the initial time moment, then the difference predictor-corrector scheme on adaptive grid preserves the state of rest at all subsequent time moments when the newly obtained conditions are used. This is one of the advantages of the developed boundary conditions in comparison with the known shock-capturing methods, where the preservation of the state of rest is usually problematic for the run-up problems. The numerical experiments have shown that for the run-up problems the substitution of a slope by a vertical wall in the initial position of the waterfront point leads to the significant change of the wave amplification in the case of very smooth slopes even if a wall embedding is small. It is expected that the obtained results will be used for solving two-dimensional problems in the framework of the classical model of shallow water, as well as in the algorithms for solution of nonlinear dispersive equations.
[full text]
Keywords: Nonlinear shallow water equations, finite-difference scheme, adaptive grid, surface waves, run-up
Author(s):
Shokina Nina Yurievna
PhD.
Position: Research Scientist
Office: Medical Center University of Freiburg
Address: 79106, Germany, Freiburg, Killianstrasse, 5a
Phone Office: (49761) 270 73930
E-mail: nina.shokina@uniklinik-freiburg.de
SPIN-code: 8680-7439Khakimzyanov Gayaz Salimovich
Dr. , Professor
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: khak@ict.nsc.ru
SPIN-code: 3144-0877 References:
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Bibliography link:
Shokina N.Y., Khakimzyanov G.S. An improved adaptive grid method for one-dimensional shallow water equations // Computational technologies. 2015. V. 20. ¹ 4. P. 83-106
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