Articles

An adaptive artificial viscosity for the displacement shallow water wave equation

Expand
  • State Key Laboratory of Structural Analysis of Industrial Equipment, Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, Shenyang Province, China

Received date: 2021-09-30

  Revised date: 2021-11-24

  Online published: 2022-01-25

Abstract

The numerical oscillation problem is a difficulty for the simulation of rapidly varying shallow water surfaces which are often caused by the unsmooth uneven bottom, the moving wet-dry interface, and so on. In this paper, an adaptive artificial viscosity (AAV) is proposed and combined with the displacement shallow water wave equation (DSWWE) to establish an effective model which can accurately predict the evolution of multiple shocks effected by the uneven bottom and the wet-dry interface. The effectiveness of the proposed AAV is first illustrated by using the steady-state solution and the small perturbation analysis. Then, the action mechanism of the AAV on the shallow water waves with the uneven bottom is explained by using the Fourier theory. It is shown that the AVV can suppress the wave with the large wave number, and can also suppress the numerical oscillations for the rapidly varying bottom. Finally, four numerical examples are given, and the numerical results show that the DSWWE combined with the AAV can effectively simulate the shock waves, accurately capture the movements of wet-dry interfaces, and precisely preserve the mass.

Cite this article

Keqi YE, Yuelin ZHAO, Feng WU, Wanxie ZHONG . An adaptive artificial viscosity for the displacement shallow water wave equation[J]. Applied Mathematics and Mechanics, 2022 , 43(2) : 247 -262 . DOI: 10.1007/s10483-022-2815-7

References

[1] LIU, W., HE, S., and XU, Q. Two-dimension coupling model to simulate water flow, sediment transport and bed evolution. Hydrology Research, 48(6), 1537-1553 (2017)
[2] JIAN, W., LIANG, D., ZHANG, J., and YANG, X. Comparison between shallow water and Boussinesq models for predicting cascading dam-break flows. Natural Hazards, 83(1), 327-343 (2016)
[3] ZHANG, T., ZHAN, C., WANG, H., LIN, C., and GUO, X. A meshless artificial viscosity method for wet-dry moving interfaces problems of shallow water flow. Ocean Engineering, 236, 109447 (2021)
[4] XING, Y. High order finite volume WENO schemes for the shallow water flows through channels with irregular geometry. Journal of Computational and Applied Mathematics, 299, 229-244 (2016)
[5] MOHAMADIAN, A., LE-ROUX, D. Y., TAJRISHI, M., and MAZAHERI, K. A mass conservative scheme for simulating shallow flows over variable topographies using unstructured grid. Advances in Water Resources, 28(5), 523-539 (2005)
[6] BORTHWICK, B. R. M. A. Adaptive Q-tree Godunov-type scheme for shallow water equations. International Journal for Numerical Methods in Fluids, 35(3), 247-280 (2001)
[7] MOUSA, M. M. and MA, W. Efficient modeling of shallow water equations using method of lines and artificial viscosity. Modern Physics Letters B, 34(4), 2050051 (2020)
[8] ISSAKHOV, A. and ZHANDAULET, Y. Numerical study of dam-break fluid flow using volume of fluid (VOF) methods for different angles of inclined planes. SIMULATION: Transactions of the Society for Modeling and Simulation International, 97(11), 717-737 (2021)
[9] WANG, B., ZHANG, F. J., LIU, X., GUO, Y. K., ZHANG, J. M., and PENG, Y. Approximate analytical solution and laboratory experiments for dam-break wave tip region in triangular channels. Journal of Hydraulic Engineering, 147(10), 6021015 (2021)
[10] LIU, W., WANG, B., and GUO, Y. Numerical study of the dam-break waves and Favre waves down sloped wet rigid-bed at laboratory scale. Journal of Hydrology, 602, 126752 (2021)
[11] MARANGOZ, H. O. and ANILAN, T. Two-dimensional modeling of flood wave propagation in residential areas after a dam break with application of diffusive and dynamic wave approaches. Natural Hazards (2021) https://doi.org/10.1007/s11069-021-04953-w
[12] MATHER, K. and JINKS, J. S. Concepts and Experiments, Springer, Berlin (1996)
[13] HWANG, Y. H. A characteristic particle method for the Saint Venant equations. Computers & Fluids, 76, 58-72 (2013)
[14] MOLLS, T. and MOLLS, F. Space-time conservation method applied to Saint Venant equations. Journal of Hydraulic Engineering, 125(5), 501-508 (1998)
[15] CHUNG, W. and KANG, Y. L. Classifying river waves by the Saint Venant equations decoupled in the Laplacian frequency domain. Journal of Hydraulic Engineering, 132(7), 666-680 (2006)
[16] GUO, X. R. Analytic rogue wave-type solutions for the generalized (2+1)-dimensional Boussinesq equation. Modern Physics Letters B, 35(22), 2150380 (2021)
[17] LIU, C. and DAI, Z. Exact periodic solitary wave solutions for the (2+1)-dimensional Boussinesq equation. Journal of Mathematical Analysis & Applications, 367(2), 444-450 (2010)
[18] ZHANG, Y. and LING, Y. E. Rational and periodic wave solutions of two-dimensional Boussinesq equation. Communications in Theoretical Physics, 49(4), 815-824 (2008)
[19] PETITPAS, F., FRANQUET, E., SAUREL, R., and METAYER, O. L. A relaxation-projection method for compressible flows. Part Ⅱ: artificial heat exchanges for multiphase shocks. Journal of Computational Physics, 225(2), 2214-2248 (2007)
[20] GUO, Y., LIU, R. X., DUAN, Y. L., and LI, Y. A characteristic-based finite volume scheme for shallow water equations. Journal of Hydrodynamics, 4, 531-540 (2009)
[21] SAMPSON, J., EASTON, A., and SINGH, M. Moving boundary shallow water flow above parabolic bottom topography. ANZIAM Journal, 47, 373-387 (2006)
[22] BRUFAU, P., GARCÍA-NAVARRO, P., and VÁZQUEZ-CENDÓN, M. E. Zero mass error using unsteady wetting-drying conditions in shallow flows over dry irregular topography. International Journal for Numerical Methods in Fluids, 45(10), 1047-1082 (2010)
[23] WU, F. and ZHONG, W. X. Constrained Hamilton variational principle for shallow water problems and Zu-class symplectic algorithm. Applied Mathematics and Mechanics (English Edition), 37(1), 1-14 (2016) https://doi.org/10.1007/s10483-016-2051-9
[24] WU, F. and ZHONG, W. X. A shallow water equation based on displacement and pressure and its numerical solution. Environmental Fluid Mechanics, 17(5), 1-28 (2017)
[25] WU, F. and ZHONG, W. On displacement shallow water wave equation and symplectic solution. Computer Methods in Applied Mechanics and Engineering, 318, 431-455 (2017)
[26] LEVEQUE, R. J. Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. Journal of Computational Physics, 146(1), 346-365 (1998)
[27] ZEIFANG, J. and BECK, A. A data-driven high order sub-cell artificial viscosity for the discontinuous Galerkin spectral element method. Journal of Computational Physics, 441, 110475 (2021)
[28] LIN, S., LUO, Q., LENG, H., and SONG, J. Alternating polynomial reconstruction method for hyperbolic conservation laws. Mathematics, 16(9), 1861-1885 (2021)
[29] WU, F., YAO, Z., and ZHONG, W. Fully nonlinear (2+1)-dimensional displacement shallow water wave equation. Chinese Physics B, 26(5), 54501 (2017)
[30] VON NEUMANN, J. and RICHTMYER, R. D. A method for the numerical calculation of hydrodynamic shocks. Journal of Applied Physics, 21(3), 232 (1950)
[31] WU, F. Numerical Modeling of Water Waves Based on Displacement: Symplectic Method (in Chinese), Dalian University of Technology Press, Dalian, 78-82 (2016)
[32] WANG, Z., ZHU, J., and ZHAO, N. A new fifth-order finite difference well-balanced multi-resolution WENO scheme for solving shallow water equations. Computers and Mathematics with Applications, 80(5), 1387-1404 (2020)
[33] PENG, L. A., WSD, B., and ZHEN, G. C. High order well-balanced finite difference WENO interpolation-based schemes for shallow water equations. Computers & Fluids, 201, 104476 (2020)
[34] BO, L. H. Study on the Shallow Water Equations of High Resolution Algorithm (in Chinese), Ph. D. dissertation, Dalian University of Technology, 42-43 (2013)
Outlines

/

APS Journals | CSTAM Journals | AMS Journals | EMS Journals | ASME Journals