DFB (Divergence Form for Bed slope source term) was rigorously derived and the error of mDFB using mean water depth at the cell
face in DFB was clearly demonstrated. In addition, DFB technique turned out to be an exact method to the bed slope source term. The
existing volume/free-surface relationship to the PSC (Partially Submerged Cell) has been corrected. It was discussed that treatment for
the partially submerged edge is required to satisfy the C-property in PSC. It is expected that this study will provides a more accurate
means in analyzing the shallow water equations with the approximate Riemann solver.

1.

2. Aureli, F., Maranzoni, A., Mignosa, P., and Ziveri, C. (2008). “A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography.” Adv. Water Resour., Vol. 31, pp. 962-974.

3. Begnudelli, L., and Sanders, B. F. (2006). “Unstructured grid finite- volume algorithm for shallow-water flow and scalar transport with wetting and drying.” J. Hydraul. Eng., Vol. 132, No. 4, pp. 371-384.

4. Begnudelli, L., and Sanders, B. F. (2007). “Conservative wetting and drying methodology for quadrilateral grid finite-volume methods.” J. Hydraul. Eng., Vol. 133, No. 3, pp. 312-322.

5. Bermúdez, A., Dervieux, A., Desideri, J.-A., and Vázquez, M. E. (1998). “Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes.” Comput. Methods Appl. Mech. Engrg., Vol. 155, pp. 49-72.

6. Bermúdez, A., and Vázquez, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Computers Fluids, Vol. 23, No. 8, pp. 1049-1071.

7. Bradford, S. F., and Katopodes, N. D. (2001). “Finite volume model for nonlevel basin irrigation.” J. Irrig. Drain. Eng., Vol. 127, No. 4, pp. 216-223.

8. Bradford, S. F., and Sanders, B. F. (2002). “Finite-volume model for shallow-water flooding of arbitrary topography.” J. Hydraul. Eng., Vol. 128, No. 3, pp. 289-298.

9. Greco, M., Iervolino, M., and Leopardi, A. (2008). “Discussion of ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 676-678.

10. Hwang, S.-Y. (2013). “Finite-volume model for shallow-water flow over uneven bottom.” J. Korea Water Resour. Assoc., Vol. 46, No. 2, pp. 139-153 (in Korean).

11. Komaei, S., and Bechteler, W. (2004). “An improved, robust implicit solution for the two dimensional shallow water equations on unstructured grids.” Proc. 2nd Conf. on Fluvial Hydraulics, IAHR, Edited by Greco, M., Carravetta, A., and Della Morte, R., Balkema, Vol. 2, pp. 1065-1072.

12. Kuiry, S. N., Pramanik, K., and Sen, D. (2008). “Finite volume method for shallow water equations with improved treatment of source terms.” J. Hydraul. Eng., Vol. 134, No. 2, p. 231-242.

13. Kuiry, S. N., and Sen, D. (2009). “Closure to ‘Finite volume method for shallow water equations with improved treatment of source terms’ by Soumendra Nath Kuiry, Kiran Pramanik, and Dhrubajyoti Sen.” J. Hydraul. Eng., Vol. 135, No. 11, pp. 1017.

14. LeVeque, R. J. (2002). Finite volume methods for hyperbolic problems. Cambridge University Press.

15. Liu, X., and García, M. H. (2008). “Discussion of ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 678-679.

16.

17.

18.

19. Simões, F. J. M. (2011). “Finite volume model for two-dimensional shallow environmental flow.” J. Hydraul. Eng., Vol. 137, No. 2, pp. 173-182.

20. Song, L., Zhou, J., Li, Q., Yang, X., et al. (2011). “An unstructured finite volume model for dam-break floods with wet/dry fronts over complex topography.” Int. J. Numer. Meth. Fluids., Vol. 67, pp. 960-980.

21.

22. Valiani, A., and Begnudelli, L. (2006). “Divergence form for bed slope source term in shallow water equations.” J. Hydraul. Eng., Vol. 132, No. 7, pp. 652-665.

23. Valiani, A., and Begnudelli, L. (2008). “Closure to ‘Divergence form for bed slope source term in shallow water equations’ by Alessandro Valiani and Lorenzo Begnudelli.” J. Hydraul. Eng., Vol. 134, No. 5, pp. 680-682.

24. Valiani, A., and Begnudelli, L. (2009). “Discussion of ‘Finite volume method for shallow water equations with improved treatment of source terms’ by Soumendra Nath Kuiry, Kiran Pramanik, and Dhrubajyoti Sen.” J. Hydraul. Eng., Vol. 135, No. 11, pp. 1016-1017.

25. Valiani, A., Caleffi, V., and Zanni, A. (2002). “Case study: Malpasset dam-break simulation using a two-dimensional finite volume method.” J. Hydraul. Eng., Vo. 128, No. 5, pp. 460-472.

26. van Leer, B. (1979). “Towards the ultimate conservative difference scheme V. A second order sequel to Godunov’s method.” J. Comput. Phys., Vol. 32, pp. 101-136.

27. Vázquez-Cendón, M. E. (1999). “Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry.” J. Comput. Phys., Vol. 148, pp. 497-526.

28.

29. Zhao, D. H., Shen, H. W., Tabios III, G. Q., Lai, J. S., et al. (1994). “Finite-volume two-dimensional unsteady-flow model for river basins.” J. Hydraul. Eng., Vol. 120, No. 7, pp. 863-883.

30. Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow-water equations.” J. Comput. Phys., Vol. 168, pp. 1-25.