This study describes the new analytical solutions of source terms (Coriolis force and bottom topography) in two-dimensional nonlinear shallow-water type equations based on fractional step-method of characteristics (FSMoC). In fact, there are two types of fractional step (FS) methods, which dictate how the equations are exploited in the system: directional splitting and time-splitting algorithm. The directional splitting method is used to split the two dimensional (2D) partial differential problems (PDEs) into sequential augmented two one dimensional (1D) problems (PDEs or ODEs). The 2D shallow-water type equations are split into two directions: x-direction and y-direction. Time splitting algorithm splits the equations into two phases: advection phase and non-advection phase. Non-advection phase deals source terms only.
Among the sequential problems, the advection phases are solved by the method of characteristics (MoC) and the non-advection phases are solved either by analytical methods or the Runge-Kutta method. 1D schemes are the bases of FS method for the construction of 2D schemes. In MoC, PDEs of shallow-water type equations model are transformed to ordinary differential equations (ODEs) and rearranged it with Riemann invariants. ODEs are applied successively integrating in every direction at constant quantities along the characteristics curves.
Basically, two types of solution methods for FSMoC are described here, which deal how the source terms are participated in the execution steps: the method 4-Step Fractional Step-Numerical Method (4FNM) uses only advection phases with/without source term; the other 5-Step Fractional Step-Semi Numerical Method (5FSNM), by giving source terms to an extra non-advection phase placed between the advection phases while executing the model. An alternative 4FNM is introduced to deal with complicated source terms. When the extra source term e.g friction term is unlikely to solve analytically, numerical schemes are necessary to carry on. The successful test of these new approaches on shallow-water type equations, along with the extension of these methods to others 3D models, seems to fit into that scenario.
Since characteristic curves for the flow do not fall on a rectangular grid system, invariants of the equations are interpolated at each time step. The Riemann invariants as well as other derivatives of the equations are interpolated at each time step along the characteristics using modified form of cubic spline interpolation. Attention has been focused on the matching performance of shallow-water type equations in the model. The successful applications of analytical/numerical solutions of source terms in the model are the main contribution of this study. The results of wave propagation, dispersion and the velocity vector of the proposed model are analyzed and discussed.

Abbott, M.B. (1966). An introduction to the method of characteristics: American Elsevier.
Abd-el-Malek, M.B., & Helal, M.M. (2009). Application of a fractional steps method for the numerical solution of the two-dimensional modeling of the Lake Mariut. Applied Mathematical Modelling, 33(2), 822-834.
Abd-el-Malek, M.B., & Helal, M.M. (2010). Application of a fractional step method for the numerical solution of the shallow water waves in a rotating rectangular basin. International Journal of Computer Mathematics, 87(10), 2268-2280.
Abiola, AA, & Nikaloaos, DK. (1988). Model for flood propagation on initially dry land. ASCE Journal of Hydraulic Engineering, 114(7), 689-707.
Akoh, R., Ii, S., & Xiao, F. (2008). A CIP/multi moment finite volume method for shallow water equations with source terms. International Journal for Numerical Methods in Fluids, 56(12), 2245-2270.
Amein, M. (1966). Streamflow routing on computer by characteristics. Water Resources Research, 2(1), 123-130.
Ancey, C., Iverson, RM, Rentschler, M., & Denlinger, RP. (2008). An exact solution for ideal dam-break floods on steep slopes. Water Resour. Res, 44, 257-362.
Bagchi, B., Das, S., & Ganguly, A. (2010). New exact solutions of a generalized shallow water wave equation. Physica Scripta, 82(2), -. doi: Artn 025003 Doi 10.1088/0031-8949/82/02/025003
Ball, F.K. (1964). An exact theory of siinple finite shallow water oscillations on a rotatirig earth. Hydraulics and Fluid Mechanics (ed. R. Silvester), 293–305.
Ball, FK. (1963). Some general theorems concerning the finite motion of a shallow rotating liquid lying on a paraboloid. Journal of Fluid Mechanics, 17(02), 240-256.
Ball, FK. (1965). The effect of rotation on the simpler modes of motion of a liquid in an elliptic paraboloid. Journal of Fluid Mechanics, 22(03), 529-545.
Bates, JR, & McDonald, A. (1982). Multiply-upstream, semi-Lagrangian advective schemes- Analysis and application to a multi-level primitive equation model. Monthly Weather Review, 110, 1831-1842.
Bates, P. D., Dawson, R. J., Hall, J. W., Matthew, S. H. F., Nicholls, R. J., Wicks, J., & Hassan, M. A. A. M. (2005). Simplified two-dimensional numerical modelling of coastal flooding and example applications. Coastal Engineering, 52(9), 793-810. doi: DOI 10.1016/j.coastaleng.2005.06.001
Bellos, CV, Soulis, JV, & Sakkas, JG. (1991). Computation of two-dimensional dam-break-induced flows. Advances in Water Resources, 14(1), 31-41.
Carrier, G.F., Wu, T.T., & Yeh, H. (2003). Tsunami run-up and draw-down on a plane beach. Journal of Fluid Mechanics, 475(-1), 79-99.
Carrier, GF, & Greenspan, HP. (1958). Water waves of finite amplitude on a sloping beach. Journal of Fluid Mechanics, 4(01), 97-109.
Cecchi, M.M., Pica, A., & Secco, E. (1998). A projection method for shallow water equations. International Journal for Numerical Methods in Fluids, 27(1 4), 81-95.
Chang, W., Giraldo, F., & Perot, B. (2002). Analysis of an exact fractional step method. Journal of Computational Physics, 180(1), 183-199.
CHEN, J., & SHI, Z. (2006). SOLUTION OF 2D SHALLOW WATER EQUATIONS BY GENUINELY MULTIDIMENSIONAL SEMI-DISCRETE CENTRAL SCHEME*. Journal of Hydrodynamics, Ser. B, 18(4), 436-442.
Choi, B.H., Pelinovsky, E., Kim, DC, Didenkulova, I., & Woo, SB. (2008). Two-and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Processes in Geophysics, 15(3), 489-502.
Choi, BH, Kaistrenko, V., Kim, KO, Min, BI, & Pelinovsky, E. (2011). Rapid forecasting of tsunami runup heights from 2-D numerical simulations. Nat. Hazards Earth Syst. Sci, 11, 707-714.
Chun, SJ, & Merkley, GP. (2008). ODE solution to the characteristic form of the Saint-Venant equations. Irrigation Science, 26(3), 213-222.
Courant, R., Friedrichs, K., & Lewy, H. (1928). Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen, 100(1), 32-74.
Courant, R., & Hilbert, D. (1962). Methods of mathematical physics, vol. 11. New York: Interscience.
Crandall, M., & Majda, A. (1980). The method of fractional steps for conservation laws. Numerische Mathematik, 34(3), 285-314.
CUSHMAN ROISIN, B. (1987). Exact analytical solutions for elliptical vortices of the shallow water equations. Tellus A, 39(3), 235-244.
Dotsenko, S., & Rubino, A. (2006). Analytical solutions for circular stratified eddies of the reduced-gravity shallow-water equations. Journal of Physical Oceanography, 36(9), 1693-1702.
Dotsenko, SF, & Rubino, A. (2003). Exact analytic solutions of the nonlinear long-wave equations in the case of axisymmetric fluid vibrations in a parabolic rotating vessel. Fluid Dynamics, 38(2), 303-309.
DUAN, Y., & LIU, R. (2007). Lattice Boltzmann simulations of triagular cavity flow and free-surface problems*. Journal of Hydrodynamics, Ser. B, 19(2), 127-134.
Elhanafy, H., & Copeland, G.J.M. (2007). Modified Method of Characteristics For The Shallow Water Equations.
Erbes, G. (1993). A semi-Lagrangian method of characteristics for the shallow-water equations. Monthly Weather Review, 121(12), 3443-3452.
Garc a-Navarro, P., Brufau, P., Burguete, J., & Murillo, J. (2008). The shallow water equations: An example of hyperbolic system.
Garcia, R., & Kahawita, R.A. (1986). Numerical solution of the St. Venant equations with the MacCormack finite difference scheme. International Journal for Numerical Methods in Fluids, 6(5), 259-274.
George, D.L. (2008). Augmented Riemann solvers for the shallow water equations over variable topography with steady states and inundation. Journal of Computational Physics, 227(6), 3089-3113.
Goldberg, D.E., & Wylie, E.B. (1983). Characteristics method using time-line interpolations. Journal of Hydraulic Engineering, 109(5), 670-683.
Guard, PA, Baldock, T., & Nielsen, P. (2011). General solutions for the initial run-up of a breaking tsunami front.
Guo, Yan, Liu, Ru-xun, Duan, Ya-li, & Li, Yuan. (2009). A characteristic-based finite volume scheme for shallow water equations. [doi: DOI: 10.1016/S1001-6058(08)60181-X]. Journal of Hydrodynamics, Ser. B, 21(4), 531-540.
Hanert, Emmanuel, Roux, Daniel Y. Le, Legat, Vincent, & Deleersnijder, Eric. (2005). An efficient Eulerian finite element method for the shallow water equations. [doi: 10.1016/j.ocemod.2004.06.006]. Ocean Modelling, 10(1-2), 115-136.
Hu, K., Mingham, C.G., & Causon, D.M. (2000). Numerical simulation of wave overtopping of coastal structures using the non-linear shallow water equations. Coastal Engineering, 41(4), 433-465.
Imai, Y., Aoki, T., & Shoucri, M. (2007). Comparison of Efficient Explicit Schemes for Shallow-Water Equations-Characteristics-Based Fractional-Step Method and Multimoment Eulerian Scheme. Journal of applied meteorology and climatology, 46(3), 388-395.
Jiwen, W., & Ruxun, L. (2001). The composite finite volume method on unstructured meshes for the two dimensional shallow water equations. International Journal for Numerical Methods in Fluids, 37(8), 933-949.
Lai, C. (1965). Flow of homogeneous density in tidal reaches, solution by the method of characteristics. Open-file Report, U.S. Geological Survey, Denver, CO,, 65–93.
Lamb, H. (1945). Hydrodynamics (6th ed.): Dover Publications, New York.
Läuter, M., Handorf, D., & Dethloff, K. (2005). Unsteady analytical solutions of the spherical shallow water equations. Journal of Computational Physics, 210(2), 535-553.
Liu, P.L.F., Lynett, P., & Synolakis, C.E. (2003). Analytical solutions for forced long waves on a sloping beach. Journal of Fluid Mechanics, 478(-1), 101-109.
Lynch, D.R., & Gray, W.G. (1979). A wave equation model for finite element tidal computations. Computers & Fluids, 7(3), 207-228.
Marchandise, E., & Remacle, J.F. (2006). A stabilized finite element method using a discontinuous level set approach for solving two phase incompressible flows. Journal of Computational Physics, 219(2), 780-800.
Matsoukis, PFC. (1992). The application of the method of characteristics for the simulation of nearly horizontal flow in two and three spatial dimensions. International Journal for Numerical Methods in Fluids, 14(4), 379-401.
Mohammadian, A., Le Roux, D. Y., & Tajrishi, M. (2007). A conservative extension of the method of characteristics for 1-D shallow flows. [doi: DOI: 10.1016/j.apm.2005.11.018]. Applied Mathematical Modelling, 31(2), 332-348.
Murawski, K. (2002). Analytical and numerical methods for wave propagation in fluid media (Vol. 7): World Scientific Pub Co Inc.
Nakamura, T., Tanaka, R., Yabe, T., & Takizawa, K. (2001). Exactly conservative semi-Lagrangian scheme for multi-dimensional hyperbolic equations with directional splitting technique. Journal of Computational Physics, 174(1), 171-207.
Ogata, Y., & Yabe, T. (2004). Multi-dimensional semi-Lagrangian characteristic approach to the shallow water equations by the CIP method. Int. J. Comput. Eng. Sci, 5(3), 699-730.
PAN, C., DAI, S., & CHEN, S. (2006). NUMERICAL SIMULATION FOR 2D SHALLOW WATER EQUATIONS BY USING GODUNOV-TYPE SCHEME WITH UNSTRUCTURED MESH*. Journal of Hydrodynamics, Ser. B, 18(4), 475-480.
Peng, X. D., Chang, Y., Li, X. L., & Xiao, F. (2010). Application of the characteristic CIP method to a shallow water model on the sphere. Advances in Atmospheric Sciences, 27(4), 728-740. doi: 10.1007/s00376-009-9148-6
Peng, Y., Zhou, J. G., & Burrows, R. (2011). Modelling solute transport in shallow water with the lattice Boltzmann method. [doi: 10.1016/j.compfluid.2011.07.008]. Computers & Fluids, 50(1), 181-188.
Perot, J.B. (1993). An analysis of the fractional step method. Journal of Computational Physics, 108(1), 51-58.
Pulch, R., & Gunther, M. (2002). A method of characteristics for solving multirate partial differential equations in radio frequency application. Applied numerical mathematics, 42(1-3), 397-409.
Rashidi, M. M., Ganji, D. D., & Dinarvand, S. (2008). Approximate traveling wave solutions of coupled Whitham-Broer-Kaup shallow water equations by homotopy analysis method. Differential Equations and Nonlinear Mechanics, 2008.
Ritter, A. (1892). Die Fortpflanzung de Wasserwellen. Zeitschrift Verein Deutscher Ingenieure, 36 (33), 947-954.
Rosatti, G., & Begnudelli, L. (2010). The Riemann Problem for the one-dimensional, free-surface Shallow Water Equations with a bed step: Theoretical analysis and numerical simulations. Journal of Computational Physics, 229(3), 760-787.
Rubino, Angelo, Brandt, Peter, & Hessner, Katrin. (1998). Analytical Solutions for Circular Eddies of the Reduced-Gravity, Shallow-Water Equations. Journal of Physical Oceanography, 28(5), 999-1002. doi: 10.1175/1520-0485(1998)028<0999:ASFCEO>2.0.CO;2
Salmon, R. (2002). Numerical solution of the two-layer shallow water equations with bottom topography. Journal of marine research, 60(4), 605-638.
Shen, MC, & Meyer, RE. (1963). Climb of a bore on a beach Part 3. Run-up. Journal of Fluid Mechanics, 16(01), 113-125.
SHI, Z., YAN, Y., YANG, F., QIAN, Y., & HU, G. (2008). A lattice boltzmann method for simulation of a three-dimensional drop impact on a liquid film*. Journal of Hydrodynamics, Ser. B, 20(3), 267-272.
Shoucri, M. (2007). Numerical solution of the shallow water equations with a fractional step method. Computer Physics Communications, 176(1), 23-32.
SLOFF, CJ. (1994). APPLICATION OF THE METHOD OF CHARACTERISTICS TO A 2-DH 2-LAYER MODEL FOR SEDIMENTATION IN RESERVOIRS. Euromech 310. Proc. Euromech conf., 152-166.
Socolofsky, S.A., & Jirka, G.H. (2005). Special topics in mixing and transport processes in the environment. Engineering–Lectures. 5th Edition. Texas A&M University, 1-93.
Sun, JS, Wai, OWH, Chau, KT, & Wong, RHC. (2009). Prediction of tsunami propagation in the Pearl River estuary. Science of Tsunami Hazards, 28(2), 142.
Takizawa, K., Yabe, T., & Nakamura, T. (2002). Multi-dimensional semi-Lagrangian scheme that guarantees exact conservation. Computer Physics Communications, 148(2), 137-159.
Tanguay, M., Robert, A., & Laprise, R. (1990). A semi-implicit semi-Lagrangian fully compressible regional forecast model. Monthly Weather Review, 118(10), 1970-1980.
Tanguay, Monique, Robert, André, & Laprise, René. (1990). A Semi-implicit Send-Lagrangian Fully Compressible Regional Forecast Model. Monthly Weather Review, 118(10), 1970-1980. doi: 10.1175/1520-0493(1990)118<1970:ASISLF>2.0.CO;2
Thacker, W.C. (1981). Some exact solutions to the nonlinear shallow-water wave equations. Journal of Fluid Mechanics, 107(-1), 499-508.
Toda, K., Ogata, Y., & Yabe, T. (2009). Multi-dimensional conservative semi-Lagrangian method of characteristics CIP for the shallow water equations. Journal of Computational Physics, 228(13), 4917-4944.
Tsai, T.L., Chiang, S.W., & Yang, J.C. (2004). Characteristics method with cubic spline interpolation for open channel flow computation. International Journal for Numerical Methods in Fluids, 46(6), 663-683.
Tsai, T.L., & Yang, J.C. (2005). Kinematic wave modeling of overland flow using characteristics method with cubic-spline interpolation. Advances in Water Resources, 28(7), 661-670.
Utsumi, T., Kunugi, T., & Aoki, T. (1997). Stability and accuracy of the cubic interpolated propagation scheme. Computer Physics Communications, 101(1-2), 9-20.
Vardy, AE. (1977). On the Use of the Method of Characteristics for the Solution of Unsteady Flows in Networks. Paper presented at the Proceedings of 2nd International Conference on Pressure Surges, British Hydromech. Research Association, Fluid Engineering, Cranfield, England.
Wang, H., & Yeh, G.T. (2005). A Characteristic-Based Semi-Lagrangian Method for Hyperbolic Systems of Conservation Laws. Chinese Journal of Atmospheric Sciences, 29(1), 21-42.
Wiggert, D.C., & Sundquist, M.J. (1977). Fixed-grid characteristics for pipeline transients. Journal of the Hydraulics division, 103(12), 1403-1416.
Wylie, E.B. (1980). Inaccuracies in the characteristics method. Paper presented at the Proc. 28th Annual Hydraulic Spec. Conf, Chicago, IL.
Yabe, T., Ishikawa, T., Wang, PY, Aoki, T., Kadota, Y., & Ikeda, F. (1991). A universal solver for hyperbolic-equations by cubic-polynomial interpolation. 2. 2-dimensional and 3-dimensional solvers. Comput. Phys. Commun, 66, 233.
Yabe, T., & Ogata, Y. (2010). Conservative semi-Lagrangian CIP technique for the shallow water equations. Computational Mechanics, 46(1), 125-134.
Yanenko, N.N. (1971). The method of fractional steps: the solution of problems of mathematical physics in several variables. Berlin: Springer-Verlag.
Yang, J.C., & Chiu, K.P. (1993). Use of characteristics method with cubic interpolation for unsteady flow computation. International Journal for Numerical Methods in Fluids, 16(4), 329-345.
Zahibo, N., Pelinovsky, E., Golinko, V., & Osipenko, N. (2005). Tsunami wave runup on coasts of narrow bays. Arxiv preprint nlin/0502020.
Zhang, X., Long, W., Xie, H., Zhu, J., & Wang, J. (2007). Numerical simulation of flood inundation processes by 2D shallow water equations. Frontiers of Architecture and Civil Engineering in China, 1(1), 107-113.
ZHU, L., & TIAN, Z. (2008). Numerical simulation of two-dimensional dam-break. Chinese Journal of Hydrodynamics.