Analysis of floodplain inundation using 2D nonlinear diffusive wave equation solved with splitting technique

• Autorzy: Gąsiorowski D.
• Języki publikacji: EN

Abstrakty

EN

In the paper a solution of two-dimensional (2D) nonlinear diffusive wave equation in a partially dry and wet domain is considered. The splitting technique which allows to reduce 2D problem into the sequence of one-dimensional (1D) problems is applied. The obtained 1D equations with regard to x and y are spatially discretized using the modified finite element method with the linear shape functions. The applied modification referring to the procedure of spatial integration leads to a more general algorithm involving a weighting parameter. Time integration is carried out using a two-level difference scheme with the weighting parameter as well. The resulting tri-diagonal systems of nonlinear algebraic equations are solved using the Picard iterative method. For particular sets of the weighting parameters, the proposed method takes the form of a standard finite element method and various schemes of the finite difference method. On the other hand, for the linear version of the governing equation, the proper values of the weighting parameters ensure an approximation of 3rd order. Since the diffusive wave equation can be solved no matter whether the area is dry or wet, the numerical computations can be carried out over entire domain of solution without distinguishing a current position of the shoreline which is obtained as a result of solution.

Słowa kluczowe

EN

unsteady surface flow; nonlinear diffusive wave equation; splitting method; finite element method; finite difference method

• Wydawca: Instytut Geofizyki PAN ; Springer
• Czasopismo: Acta Geophysica
• Rocznik: 2013
• Tom: Vol. 61, no. 3
• Strony: 668--689
• Opis fizyczny: Bibliogr. 33 poz.

Twórcy

autor

Gąsiorowski D.

• Faculty of Civil and Environmental Engineering, Gda ń sk University of Technology, Gdańsk, Poland

Bibliografia

• 1. Ascher, U.M., and L.R. Petzold (1998), Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia.
• 2. Bates, P.D., and M.S. Horritt (2005), Modelling wetting and drying processes in hydraulic models. In: P.D. Bates, S.N. Lane, and R.I. Ferguson (eds.), Computational Fluid Dynamics: Applications in Environmental Hydraulics, John Wiley & Sons Ltd., New York, 121-146, DOI: 10.1002/0470015195. ch6.
• 3. Cea, L., M. Garrido, and J. Puertas (2010), Experimental validation of twodimensional depth-averaged models for forecasting rainfall-runoff from precipitation data in urban areas, J. Hydrol. 382,1-4, 88-102, DOI: 10.1016/j.jhydrol.2009.12.020.
• 4. Cunge, J.A. (1975), Two dimensional modeling of flood plains. In: K. Mahmood and V. Yevjevich (eds.), Unsteady Flow in Open Channels, Vol. 2, Water Resources Publs., Fort Collins, 705-762.
• 5. Dahlquist, G., and A. Björck (1974), Numerical Methods, Prentice-Hall, Englewood Cliffs.
• 6. Fletcher, C.A.J. (1991), Computational Techniques for Fluid Dynamics, Vol. 1, Springer, New York.
• 7. Gresho, P.M., and R.L. Sani (1998), Incompressible Flow and the Finite Element Method. Volume 1: Advection-Diffusion and Isothermal Laminar Flow, John Wiley & Sons Inc., New York.
• 8. Heniche, M., Y. Secretan, P. Boudreau, and M. Leclerc (2000), A two-dimensional finite element drying-wetting shallow water model for rivers and estuaries, Adv. Water Resour. 23,4, 359-372, DOI: 10.1016/S0309-1708(99)00031-7.
• 9. Hervouet, J.-M. (2007), Hydrodynamics of Free Surface Flows - Modelling with the Finite Element Method, John Wiley & Sons Ltd., Chichester, 360 pp.
• 10. Horritt, M.S. (2002), Evaluating wetting and drying algorithms for finite element models of shallow water flow, Int. J. Numer. Meth. Eng. 55,7, 835-851, DOI: 10.1002/nme.529.
• 11. Horritt, M.S., and P.D. Bates (2001), Predicting floodplain inundation: raster-based modelling versus the finite-element approach, Hydrol. Process. 15,5, 825-842, DOI: 10.1002/hyp.188.
• 12. Hromadka, T.V., and C.C. Yen (1986), A diffusion hydrodynamic model (DHM), Adv. Water Resour. 9,3, 118-170, DOI: 10.1016/0309-1708(86)90031-X.
• 13. Hsu, M.H., S.H. Chen, and T.J. Chang (2000), Inundation simulation for urban drainage basin with storm sewer system, J. Hydrol. 234,1-2, 21-37, DOI: 10.1016/S0022-1694(00)00237-7.
• 14. Jothiprasad, G., and D.A. Caughey (2008), A collocated, iterative fractional-step method for incompressible large eddy simulation, Int. J. Numer. Meth. Fluids 58,4, 355-380, DOI: 10.1002/fld.1713.
• 15. Kalinowska, M.B., P.M. Rowiński, J. Kubrak, and D. Mirosøaw-Świątek (2012), Scenarios of the spread of a waste heat discharge in a river - Vistula River case study, Acta Geophys. 60,1, 214-231, DOI: 10.2478/s11600-011-0045-x.
• 16. LeVeque, R.J. (2002), Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambrige University Press, Cambridge, 578 pp.
• 17. Liang, Q., and A.G.L. Borthwick (2009), Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography, Comput. Fluids 38,2, 221-234, DOI: 10.1016/j.compfluid.2008.02.008.
• 18. Louaked, M., and L. Hanich (1998), TVD scheme for the shallow water equations, J. Hydraul. Res. 36,3, 363-378, DOI: 10.1080/00221689809498624.
• 19. Morita, M., and B.C. Yen (2000), Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows, Int. J. Numer. Meth. Fluids 32,8, 921-957, DOI: 10.1002/(SICI)1097-0363(20000430)32:8〈921::AID-FLD993〉3.0.CO;2-3.
• 20. Moussa, R., and C. Bocquillon (2009), On the use of the diffusive wave for modelling extreme flood events with overbank flow in the floodplain, J. Hydrol. 374,1-2, 116-135, DOI: 10.1016/j.jhydrol.2009.06.006.
• 21. Potter, D. (1973), Computational Physics, John Wiley & Sons Ltd., New York.
• 22. Press, W.H., S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery (1992), Numerical Recipes in C, the Art of Scientific Computing, 2nd ed., Cambridge University Press, Cambridge.
• 23. Prestininzi, P. (2008), Suitability of the diffusive model for dam break simulation: Application to a CADAM experiment, J. Hydrol. 361,1-2, 172-185, DOI: 10.1016/j.jhydrol.2008.07.050.
• 24. Singh, V.P. (1996), Kinematic Wave Modeling in Water Resources: Surface-Water Hydrology, John Wiley & Sons Ltd., New York, 1420 pp.
• 25. Szydłowski, M. (2001), Two-dimensional shallow water model rapidly and gradually varied flow, Arch. Hydro-Eng. Environ. Mech. 48,1, 35-61.
• 26. Szydłowski, M. (2008), Two-dimensional diffusion wave model for numerical simulation of inundation - Upper Narew case study, Publs. Inst. Geophys. Pol. Acad. Sc. E-9,405, 107-120.
• 27. Szymkiewicz, R. (1995), Method to solve 1D unsteady transport and flow equations, J. Hydraul. Eng. ASCE 121,5, 396-403, DOI: 10.1061/(ASCE)0733-9429(1995)121:5(396).
• 28. Szymkiewicz, R. (2010), Numerical Modeling in Open Channel Hydraulics, Water Science and Technology Library, Vol. 83, Springer, Dordrecht, DOI: 10.1007/978-90-481-3674-2.
• 29. Szymkiewicz, R., and D. Gąsiorowski (2012), Simulation of unsteady flow over floodplain using the diffusive wave equation and the modified finite element method, J. Hydrol. 464-465, 165-175, DOI: 10.1016/j.jhydrol.2012.07.009.
• 30. Tan, W. (1992), Shallow Water Hydrodynamics, Elsevier Oceanography Series, Vol. 55, Elsevier, Amsterdam, 434 pp.
• 31. Weill, S., E. Mouche, and J. Patin (2009), A generalized Richards equation for surface/subsurface flow modelling, J. Hydrol. 366,1-4, 9-20, DOI: 10.1016/j.jhydrol.2008.12.007.
• 32. Zhang, X.H., H. Ishidaira, K. Takeuchi, Z.X. Xu, and X.W. Zhang (2004), An approach to inundation simulation in large river basins using the triangle finite difference method, J. Environ. Inform. 3,1, 51-63, DOI: 10.3808/jei.200400027.
• 33. Zienkiewicz, O.C. (1971), The Finite Element Method in Engineering Science, McGraw-Hill, London.