## “Third Order Reconstruction of the KP Scheme for Model of River Tinnelva”Authors: Susantha Dissanayake, Roshan Sharma and Bernt Lie,
Affiliation: University College of Southeast Norway
Reference: 2017, Vol 38, No 1, pp. 33-50. |

**Keywords:**Run-of-river hydropower, Saint Venant Equations, KP scheme, CWENO

**Abstract:**The Saint-Venant equation/Shallow Water Equation is used to simulate flow of river, flow of liquid in an open channel, tsunami etc. The Kurganov-Petrova (KP) scheme which was developed based on the local speed of discontinuity propagation, can be used to solve hyperbolic type partial differential equations (PDEs), hence can be used to solve the Saint-Venant equation. The KP scheme is semi discrete: PDEs are discretized in the spatial domain, resulting in a set of Ordinary Differential Equations (ODEs). In this study, the common 2nd order KP scheme is extended into 3rd order scheme while following the Weighted Essentially Non-Oscillatory (WENO) and Central WENO (CWENO) reconstruction steps. Both the 2nd order and 3rd order schemes have been used in simulation in order to check the suitability of the KP schemes to solve hyperbolic type PDEs. The simulation results indicated that the 3rd order KP scheme shows some better stability compared to the 2nd order scheme. Computational time for the 3rd order KP scheme for variable step-length ode solvers in MATLAB is less compared to the computational time of the 2nd order KP scheme. In addition, it was confirmed that the order of the time integrators essentially should be lower compared to the order of the spatial discretization. However, for computation of abrupt step changes, the 2nd order KP scheme shows a more accurate solution.

PDF (1616 Kb) DOI: 10.4173/mic.2017.1.4

**References:**

[1] Arshed, G.M. and Hoffmann, K.A. (2013). Arshed, G, M. and Hoffmann, K.A. Minimizing errors from linear and nonlinear weights of weno scheme for broadband applications with shock waves. Journal of Computational Physics. 246:58--77. doi:10.1016/j.jcp.2013.03.037

[2] Dissanayake, S., Sharma, R., and Lie, B. (2016). Dissanayake, S, , Sharma, R., and Lie, B. Semi discrete scheme for the solution of flow in river tinnelva. In Proceedings of EUROSIM 2016. IEEE, Oulu, Finland, pages 134--139. doi:10.1109/EUROSIM.2016.142

[3] Fayssal, B., Saida, S., and Mohammed, S. (2015). Fayssal, B, , Saida, S., and Mohammed, S. Projection finite volume method for shallow water flows. Mathematics and Computers in Simulation. 118:87--101. doi:10.1016/j.matcom.2014.11.027

[4] Griffiths, D.F., Dold, J.W., and Silvester, D.J. (2015). Griffiths, D, F., Dold, J.W., and Silvester, D.J. Essential Partial Differential Equations Analytical and Computational Aspects. Switzerland: Springer International Publishing. doi:10.1007/978-3-319-22569-2

[5] Harten, A., Engquist, B., Osher, S., and Chakravarthy, S. (1987). Harten, A, , Engquist, B., Osher, S., and Chakravarthy, S. Uniformly high order essentially nonoscillatory schemes, iii. Journal of Computational Physics. 71:231--303. doi:10.1006/jcph.1996.5632

[6] Hawary, S.E., Abu-Elyazeed, O. S.M., Fahmy, A.A., and Meglaa, K. (2016). Hawary, S, E., Abu-Elyazeed, O. S.M., Fahmy, A.A., and Meglaa, K. Theoretical study of hydraulic jump during circular horizontal hot leg injection in pressurized water reactor. Annals of Nuclear Energy. 94:783--792. doi:10.1016/j.anucene.2016.04.040

[7] Kurganov, A. and Levy, D. (2000). Kurganov, A, and Levy, D. A third-order semidiscrete central scheme for conservation laws and convection diffusion equations. SIAM Journal on Scientific Computing. 22(4):1461--1488. doi:10.1137/S1064827599360236

[8] Kurganov, A. and Petrova, G. (2007). Kurganov, A, and Petrova, G. A second order well-balanced positivity preserving central-upwind scheme for the saint-venant system. Communications in Mathematical Science. 5(1):133--160. doi:10.4310/CMS.2007.v5.n1.a6

[9] Kurganov, A. and Tadmor, E. (1999). Kurganov, A, and Tadmor, E. New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of Computational Physics. 160:241--282. doi:10.1006/jcph.2000.6459

[10] LeVeque, R.J. (1999). LeVeque, R, J. Numerical method for conservation Laws. Boston: Birkhäuser Verlag -Basel, Switzerland, second edition. doi:10.1007/978-3-0348-8629-1

[11] Levy, D., Pupppo, G., and Russo, G. (2000). Levy, D, , Pupppo, G., and Russo, G. Compact central weno schemes for multidimensional conservation laws. SIAM Journal on Scientific Computing. 22:656--672. doi:10.1137/S1064827599359461

[12] xia Li, L., sheng Liao, H., Liu, D., and yin Jiang, S. (2015). xia Li, L, , sheng Liao, H., Liu, D., and yin Jiang, S. Experimental investigation of the optimization of stilling basin with shallow-water cushion used for low froude number energy dissipation. Journal of Hydrodynamics. 27:522--529. doi:10.1016/S1001-6058(15)60512-1

[13] Liu, X.-D., Osher, S., and Chan, T. (1994). Liu, X, -D., Osher, S., and Chan, T. Weighted essentially non-oscillatory schemes. Journal of Computational Physics. 115:200--212. doi:10.1006/jcph.1994.1187

[14] Liu, X.-D. and Tadmor, E. (1998). Liu, X, -D. and Tadmor, E. Third order non-oscillatory central scheme for hyperbolic conservation laws. Numerische Mathematik Electronic Edition. 79:397--425. doi:10.1007/s002110050345

[15] Salame, C., Aillerie, M., Papageorgas, P., kateb, S., Debabeche, M., and Riguet, F. (2015). Salame, C, , Aillerie, M., Papageorgas, P., kateb, S., Debabeche, M., and Riguet, F. Hydraulic jump in a sloped trapezoidal channel. Energy Procedia. 74:251--257. doi:10.1016/j.egypro.2015.07.591

[16] Shampine, L.F., Gladwell, I., and Thompson, S. (2003). Shampine, L, F., Gladwell, I., and Thompson, S. Solving ODEs with MATLAB. Cambridge University Press, first edition. .

[17] Sharma, R. (2015). Sharma, R, Second order scheme for open channel flow. Technical report, Telemark Open Research Archive (TEORA), University college of Southeast Norway, Porsgrunn, Norway. http://hdl.handle.net/2282/2575, .

[18] Shi, J., Hu, C., and Shu, C.-W. (2002). Shi, J, , Hu, C., and Shu, C.-W. A technique of treating negative weights in weno schemes. Journal of Computational Physics. 175:108--127. doi:10.1006/jcph.2001.6892

[19] Shu, C.-W. (2003). Shu, C, -W. High order finite difference and finite volume weno schemes and discontinuous galerkin methods for cfd. International Journal of Computational Fluid Dynamics. 17:107--118. doi:10.1080/1061856031000104851

[20] Titarev, V.A. and Toro, E.F. (2004). Titarev, V, A. and Toro, E.F. Finite-volume weno schemes for three-dimensional conservation laws. Journal of Computational Physics. 201:238--260. doi:10.1016/j.jcp.2004.05.015

[21] Ujevic, N. (2007). Ujevic, N, New error bounds for the simpson’s quadrature rule and applications. Computers & Mathematics with Applications. 53:64--72. doi:10.1016/j.camwa.2006.12.008

[22] Versteeg, H.K. and Malalasekera, W. (2007). Versteeg, H, K. and Malalasekera, W. An introduction to computational fluid dynamics. England Pearson Education Limited, second edition. .

[23] Zhang, Y.T. and Shu, C. (2009). Zhang, Y, T. and Shu, C. Third order weno scheme on three dimensional tetrahedral meshes. Communication in computational physics. 5:836--848. .

**BibTeX:**

@article{MIC-2017-1-4,

title={{Third Order Reconstruction of the KP Scheme for Model of River Tinnelva}},

author={Dissanayake, Susantha and Sharma, Roshan and Lie, Bernt},

journal={Modeling, Identification and Control},

volume={38},

number={1},

pages={33--50},

year={2017},

doi={10.4173/mic.2017.1.4},

publisher={Norwegian Society of Automatic Control}

};