STUDY THE PERFORMANCE OF THE TWO WAVELET-BASED ADAPTATION SCHEMES FOR THE SHALLOW WATER FLOW MODELLING

  • DILSHAD A. HALEEM Dept. Water Resources,College of Engineering,University of Duhok, Kurdistan Region-Iraq
  • GEORGE KESSERWANI Civil and Structural Engineering Department,University of Sheffield, UK
  • AZA HANI SHUKRI Dept. Civil Engineering,College of Engineering,University of Duhok, Kurdistan Region-Iraq
  • ALAN SAHEEN SAIFALDEEN Dept. Civil Engineering,College of Engineering,University of Duhok, Kurdistan Region-Iraq
Keywords: Finite volume method, Discontinues Galerkin method, Haar wavelets, wavelets

Abstract

This work proposed a new adaptive method which avails from the wavelets theory for transforming the local single resolution information into multiresolution information. This information became accessible and by deactivating or activating them, the spatial resolution adaptation was achieved. The adaptive technique was  combined with two standard numerical modelling schemes (i.e. finite volume and discontinuous Galerkin schemes) to produce two new adaptive schemes for modelling one dimensional shallow water flows so-called the Haar wavelets finite volume (HWFV) and multiwavelet discontinuous Galerkin (MWDG) schemes. Both adaptive schemes were tested using hydraulic test cases. The results demonstrated that the proposed adaptive technique could serve as the foundation on which to construct complete adaptive schemes for simulating the real problems of shallow water flow. 

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References

1. Bouchut, F., Efficient numerical finite volume schemes for shallow water models. Edited Series on Advances in Nonlinear Science and Complexity, 2007. 2: p. 189-256.
2. Toro, E.F., Shock-capturing methods for free-surface shallow flows. 2001: Wiley.
3. Toro, E.F. and P. Garcia-Navarro, Godunov-type methods for free-surface shallow flows: A review. Journal of Hydraulic Research, 2007. 45(6): p. 736-751.
4. Harten, A., P.D. Lax, and B.v. Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM review, 1983. 25(1): p. 35-61.
5. Fraccarollo, L., H. Capart, and Y. Zech, A Godunov method for the computation of erosional shallow water transients. International journal for Numerical Methods in fluids, 2003. 41(9): p. 951-976.
6. Kesserwani, G., Topography discretization techniques for Godunov-type shallow water numerical models: a comparative study. Journal of Hydraulic Research, 2013. 51(4): p. 351-367.
7. Bradford, S.F. and B.F. Sanders, Finite-volume model for shallow-water flooding of arbitrary topography. Journal of Hydraulic Engineering, 2002. 128(3): p. 289-298.
8. Begnudelli, L. and B.F. Sanders, Conservative wetting and drying methodology for quadrilateral grid finite-volume models. Journal of Hydraulic Engineering, 2007. 133(3): p. 312-322.
9. Garcia-Navarro, P. and M.E. Vazquez-Cendon, On numerical treatment of the source terms in the shallow water equations. Computers & Fluids, 2000. 29(8): p. 951-979.
10. Audusse, E., et al., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM Journal on Scientific Computing, 2004. 25(6): p. 2050-2065.
11. Skoula, Z., A. Borthwick, and C. Moutzouris, Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids. International Journal of Computational Fluid Dynamics, 2006. 20(9): p. 621-636.
12. Nikolos, I. and A. Delis, An unstructured node-centered finite volume scheme for shallow water flows with wet/dry fronts over complex topography. Computer Methods in Applied Mechanics and Engineering, 2009. 198(47): p. 3723-3750.
13. Hovhannisyan, N., S. Müller, and R. Schäfer, Adaptive multiresolution discontinuous galerkin schemes for conservation laws. Mathematics of Computation, 2014. 83(285): p. 113-151.
14. Kesserwani, G. and Q. Liang, Dynamically adaptive grid based discontinuous Galerkin shallow water model. Advances in Water Resources, 2012. 37: p. 23-39.
15. Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Communications on Pure and Applied Mathematics, 1995. 48(12): p. 1305-1342.
16. Müller, S., Adaptive multiscale schemes for conservation laws. Vol. 27. 2003: Springer.
17. Keinert, F., Wavelets and multiwavelets. 2003: CRC Press.
18. Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of computational physics, 1981. 43(2): p. 357-372.
19. GUF, J.-S. and W.-S. Jiang, The Haar wavelets operational matrix of integration. International Journal of Systems Science, 1996. 27(7): p. 623-628.
20. Cockburn, B. and C.-W. Shu, Runge–Kutta discontinuous Galerkin methods for convection-dominated problems. Journal of scientific computing, 2001. 16(3): p. 173-261.
21. Gerhard, N. and S. Müller, Adaptive multiresolution discontinuous Galerkin schemes for conservation laws: multi-dimensional case. Computational and Applied Mathematics, 2013: p. 1-29.
22. Bermudez, A. and Vazquez, M. E. (1994a). Upwind methods for hyperbolic conservation laws with source terms. Computers and Fluids, 23(8):1049-1071.
23. Kesserwani, G., Gerhard, N., Caviedes-Voullime, D., Haleem, D. A., and Muller, S. (2014). A multi-resolution discontinuous Galerkin method for one dimensional shallow water ow modelling. the 3rd IAHR Europe Congress with the theme Water Engineering and Research. Porto, Portugal.
24. Haleem, D. A., Kesserwani, G., and Caviedes-Voullime, D. (2015). Haar waveletbased adaptive finite volume shallow water solver. Journal of Hydroinformatics. Vol. 17(6):857-873
25. MacDonald, I. (1996). Analysis and computation of steady open channel flow. PhD. Thesis. University of Reading.
26. Burguete Tolosa, J., Garcia-Navarro, P., Murillo, J., et al. (2008). Friction term discretization and limitation to preserve stability and conservation in the 1D shallow-water model: Application to unsteady irrigation and river flow. International Journal for Numerical Methods in Fluids, 58(4):403-425.
27. Haleem, D., Wavelet-based numerical methods adaptive modelling of shallow water flows. 2015, University of Sheffield.
Published
2017-07-29
How to Cite
HALEEM, D. A., KESSERWANI, G., SHUKRI, A. H., & SAIFALDEEN, A. S. (2017). STUDY THE PERFORMANCE OF THE TWO WAVELET-BASED ADAPTATION SCHEMES FOR THE SHALLOW WATER FLOW MODELLING. Journal of Duhok University, 20(1), 716-726. https://doi.org/10.26682/sjuod.2017.20.1.62