Инд. авторы:	Шокина Н.Ю., Хакимзянов Г.С.
Заглавие:	Усовершенствованный метод адаптивных сеток для одномерных уравнений мелкой воды
Библ. ссылка:	Шокина Н.Ю., Хакимзянов Г.С. Усовершенствованный метод адаптивных сеток для одномерных уравнений мелкой воды // Вычислительные технологии. - 2015. - Т.20. - № 4. - С.83-106. - ISSN 1560-7534. - EISSN 2313-691X.
Внешние системы:	РИНЦ: 24339608;
Реферат:	rus: Представлен модифицированный алгоритм реализации явной схемы предиктор-корректор, основанный на новом способе вычисления правой части уравнений мелкой воды на неровном дне. Предложен новый способ выбора схемных параметров, гарантирующий выполнение TVD-свойства для усовершенствованной схемы предиктор-корректор. Описан способ получения консервативных схем на подвижных сетках, основанный на выборе соответствующих значений схемных параметров в схеме предиктор-корректор. На основе известных аналитических решений уравнений мелкой воды в окрестности границы вода - суша разработаны уточненные разностные краевые условия в подвижной точке уреза, аппроксимирующие аналитические решения с большей точностью, чем использованные ранее. Приведены результаты расчетов с помощью усовершенствованного метода адаптивных сеток. eng: 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.
Ключевые слова:	adaptive grid; finite-difference scheme; nonlinear shallow water equations; накат на берег; поверхностные волны; адаптивная сетка; конечно-разностная схема; нелинейные уравнения мелкой воды; run-up; surface waves;
Издано:	2015
Физ. характеристика:	с.83-106
Цитирование:	1. Shokin, Yu.I., Fedotova, Z.I., Khakimzyanov, G.S. Hierarchy of nonlinear models of the hydrodynamics of long surface waves. Doklady Physics. 2015; 60(5):224-228. 2. Toro, E., Garcia-Navarro, P. Godunov-type methods for free-surface shallow flows: A review. Journal of Hydraulic Research. 2007; 45(6):736-751. 3. Khakimzyanov, G.S., Shokina, N.Yu. Adaptive grid method for one-dimensional shallow water equations. Computational Technologies. 2013; 18(3):54-79. (In Russ.) 4. Vazquez-Cendon, M.E. Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. Journal of Computational Physics. 1999; 148(2):497-526. 5. Voltsinger, N.E., Pelinovskii, E.N., Klevannyi, K.A. Dlinnovolnovaya dinamika pribrezhnoi zony. Leningrad: Gidrometeoizdat; 1989: 272. (In Russ.) 6. LeVeque, R.J. Numerical methods for conservation laws. Berlin: Birkhauser Verlag; 1992: 214. 7. Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics: a Practical Introduction. Berlin: Springer-Verlag; 2009: 724. 8. Lynch, D.R., Gray, W.G. Finite element simulation of flow in deforming regions. Journal of Computational Physics. 1980; (36):135-153. 9. Hou, J., Liang, Q., Zhang, H., Hinkelmann, R. An efficient unstructured MUSCL scheme for solving the 2D shallow water equations. Environmental Modelling & Software. 2015; (66):131-152. 10. Hou, J., Liang, Q., Simons, F., Hinkelmann, R. A stable 2D unstructured shallow flow model for simulations of wetting and drying over rough terrains. Computers & Fluids. 2013; (82):132-147. 11. Duran, A., Marche, F. Recent advances on the discontinuous Galerkin method for shallow water equations with topography source terms. Computers & Fluids. 2014; (101): 88-104. 12. Funke, S.W., Pain, C.C., Kramer, S.C., Piggott, M.D. A wetting and drying algorithm with a combined pressure/free-surface formulation for non-hydrostatic models. Advances in Water Resources. 2011; 34(11):1483-1495. 13. Karna, T., de Brye, B., Gourgue, O., Lambrechts, J., Comblen, R., Legat, V., Deleersnijder, E. A fully implicit wetting-drying method for DG-FEM shallow water models, with an application to the Scheldt Estuary. Computer Methods in Applied Mechanics and Engineering. 2011; 200(5-8):509-524. 14. Li, Y., Raichlen, F. Non-breaking and breaking solitary wave run-up. Journal of Fluid Mechanics. 2002; (456):295-318. 15. Flouri, E.T., Kalligeris, N., Alexandrakis, G., Kampanis, N.A., Synolakis, C.E. Application of a finite difference computational model to the simulation of earthquake generated tsunamis. Applied Numerical Mathematics. 2013; (67):111-125. 16. Shokin, Yu.I., Beisel, S.A., Rychkov, A.D., Chubarov, L.B. Numerical simulation of the tsunami runup on the coast using the method of large particles. Mathematical Models and Computer Simulations. 2015; 7(4):339-348. 17. Samarskii, A.A. The Theory of Difference Schemes. USA: Marcel Dekker, Inc.; 2001: 788. 18. Khakimzyanov, G.S., Shokina, N.Yu. Some notes on monotonicity preserving schemes. Computational Technologies. 2012; 17(2):78-98. (In Russ.) 19. Shokina, N.Yu. To the problem of construction of difference schemes on movable grids. Russian Journal of Numerical Analysis and Mathematical Modelling. 2012; 27(6):603-626. 20. Bautin, S.P., Deryabin, S.L., Sommer, A.F., Khakimzyanov, G.S., Shokina, N.Yu. Use of analytic solutions in the statement of difference boundary conditions on a movable shoreline. Russian Journal of Numerical Analysis and Mathematical Modelling. 2011; 26(4):353-377. 21. Synolakis, C.E. The runup of solitary waves. Journal of Fluid Mechanics. 1987; 185:523-545. 22. Beizel, S.A., Shokina, N.Yu., Khakimzyanov, G.S., Chubarov, L.B., Kovyrkina, O.A., Ostapenko, V.V. On some numerical algorithms for computation of tsunami runup in the framework of shallow water model. I. Computational Technologies. 2014; 19(1):40-62. (In Russ.) 23. Gusev, O.I., Shokina, N.Yu., Kutergin, V.A., Khakimzyanov, G.S. Numerical modelling of surface waves generated by underwater landslide in a reservoir. Computational Technologies. 2013; 18(5):74-90. (In Russ.) 24. Shokin, Yu.I., Beisel, S.A., Gusev, O.I., Khakimzyanov, G.S., Chubarov, L.B., Shokina, N.Yu. Numerical modelling of dispersive waves generated by landslide motion. Bulletin of the South Ural State University. 2014; 7(1):121-133. (In Russ.) 25. Rozhdestvenskiy, B.L., Yanenko, N.N. Sistemy kvazilineynykh uravneniy i ikh prilozheniya k gazovoy dinamike [Systems of Quasilinear Equations and Their Application to Gas Dynamics]. Moscow: Nauka; 1978: 688. (In Russ.) 26. Pedersen, G., Gjevik, B. Run-up of solitary waves. Journal of Fluid Mechanics. 1983; (135):283-299. 27. Cooker, M.J., Weidman, P.D., Bale, D.S. Reflection of a high-amplitude solitary wave at a vertical wall. Journal of Fluid Mechanics. 1997; (342):141-158. 28. Synolakis, C.E., Bernard, E.N., Titov, V.V., Kanoglu, U., Gonzalez, F.I. Validation and verification of tsunami numerical models. Pure and Applied Geophysics. 2008; (165):2197-2228. 29. Horrillo, J., Grilli, S.T., Nicolsky, D., Roeber, V., Zhang, J. Performance benchmarking tsunami models for NTHMP’s inundation mapping activities. Pure and Applied Geophysics. 2015; 170(3-4):1333-1359.