# Well-posedness analysis and numerical implementation of a linearized two-dimensional bottom sediment transport problem

- 17 Downloads
- 1 Citations

## Abstract

A two-dimensional linearized model of coastal sediment transport due to the action of waves is studied. Up till now, one-dimensional sediment transport models have been used. The model under study makes allowance for complicated bottom relief, the porosity of the bottom sediment, the size and density of sediment particles, gravity, wave-generated shear stress, and other factors. For the corresponding initial–boundary value problem the uniqueness of a solution is proved, and an a priori estimate for the solution norm is obtained depending on integral estimates of the right-hand side, boundary conditions, and the norm of the initial condition. A conservative difference scheme with weights is constructed that approximates the continuous initial–boundary value problem. Sufficient conditions for the stability of the scheme, which impose constraints on its time step, are given. Numerical experiments for test problems of bottom sediment transport and bottom relief transformation are performed. The numerical results agree with actual physical experiments.

### Keywords

sediment transport model coastal zone bottom surface solution uniqueness estimate for the norm of the solution to an initial–boundary value problem## Preview

Unable to display preview. Download preview PDF.

### References

- 1.I. O. Leont’ev,
*Coastal Dynamics: Waves, Currents, and Sediment Flow*(GEOS, Moscow, 2001) [in Russian].Google Scholar - 2.Liu Xiaoying, Qi Shi, Huang Yuan, Chen Yuehong, and Du Pengfei, “Predictive modeling in sediment transportation across multiple spatial scales in the Jialing River basin of China,” Int. J. Sediment Res.
**30**(3), 250–255 (2015).CrossRefGoogle Scholar - 3.G. I. Marchuk, V. P. Dymnikov, and V. B. Zalesny,
*Mathematical Models in Geophysical Hydrodynamics and Numerical Methods of Their Implementation*(Gidrometeoizdat, Leningrad, 1987) [in Russian].Google Scholar - 4.A. I. Sukhinov, A. E. Chistyakov, E. F. Timofeeva, and A. V. Shishenya, “Mathematical model for calculating coastal wave processes,” Math. Models Comput. Simul.
**5**(2), 122–129 (2013).MathSciNetCrossRefGoogle Scholar - 5.A. I. Sukhinov, A. E. Chistyakov, and E. A. Protsenko, “Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs,” Math. Models Comput. Simul.
**6**(4), 351–363 (2014).MathSciNetCrossRefMATHGoogle Scholar - 6.A. I. Sukhinov, A. E. Chistyakov, and E. A. Protsenko, “Mathematical modeling of sediment transport in coastal systems on multiprocessor computers,” Vychisl. Metody Program. Novye Vychisl. Tekhnol.
**15**(4), 610–620 (2014).Google Scholar - 7.A. I. Sukhinov, E. A. Protsenko, A. E. Chistyakov, and S. A. Shreter, “Comparison of numerical efficiency of explicit and implicit schemes as applied to sediment transport in coastal systems,” Vychisl. Metody Program. Novye Vychisl. Tekhnol.
**16**(3), 328–338 (2015).Google Scholar - 8.A. V. Shishenya and A. I. Sukhinov, “Development of a regularized mathematical model for modeling hydrodynamics and surge phenomena in shallow water basins and its parallel implementation on a parallel shared memory computer system,” Izv. Yuzhn. Fed. Univ. Tekh. Nauki
**161**(12), 219–230 (2014).Google Scholar - 9.B. N. Chetverushkin, “Resolution limits of continuous media models and their mathematical formulations,” Math. Models Comput. Simul.
**5**(3), 266–279 (2013).MathSciNetCrossRefGoogle Scholar - 10.V. S. Vladimirov,
*Equations of Mathematical Physics*(Marcel Dekker, New York, 1971; Nauka, Moscow, 1981).MATHGoogle Scholar - 11.O. A. Ladyzhenskaya,
*The Boundary Value Problems of Mathematical Physics*(Nauka, Moscow, 1973; Springer- Verlag, New York, 1985).Google Scholar - 12.O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva,
*Linear and Quasilinear Equations of Parabolic Type*(Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).MATHGoogle Scholar - 13.A. A. Samarskii,
*The Theory of Difference Schemes*(Nauka, Moscow, 1983; Marcel Dekker, New York, 2001).MATHGoogle Scholar - 14.V. S. Vasil’ev and A. I. Sukhinov, “Precise two-dimensional models for shallow water basins,” Mat. Model.
**15**(10), 17–34 (2003).MathSciNetGoogle Scholar - 15.A. I. Sukhinov, “Precise fluid dynamics models and their application in prediction and reconstruction of extreme events in the Sea of Azov,” Izv. Taganrog. Radiotech. Univ., No. 3, 228–235 (2006).Google Scholar
- 16.A. I. Sukhinov and A. V. Shishenya, “Increasing efficiency of alternating triangular method based on improved spectral estimates,” Math. Models Comput. Simul.
**5**(3), 257–265 (2013).MathSciNetCrossRefGoogle Scholar - 17.A. I. Sukhinov and A. E. Chistyakov, “Adaptive modified alternating triangular iterative method for solving grid equations with a non-self-adjoint operator,” Math. Models Comput. Simul.
**4**(4), 398–409 (2012).MathSciNetCrossRefGoogle Scholar