Approximate Riemann solutions of the two-dimensional shallow-water equations. (English) Zbl 0707.76068
Summary: A finite-difference scheme based on flux difference splitting is presented for the solution of the two-dimensional shallow-water equations of ideal fluid flow. A linearized problem, analogous to that of Riemann for gasdynamics, is defined and a scheme, based on numerical characteristic decomposition, is presented for obtaining approximate solutions to the linearized problem. The method of upwind differencing is used for the resulting scalar problems, together with a flux limiter for obtaining a second-order scheme which avoids non-physical, spurious oscillations. An extension to the two-dimensional equations with source terms, is included. The scheme is applied to a dam-break problem with cylindrical symmetry.
MSC:
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |
Keywords:
finite-difference scheme; flux difference splitting; two-dimensional shallow-water equations; ideal fluid flow; method of upwind differencing; second-order scheme; two-dimensional equations; dam-break problem with cylindrical symmetryReferences:
| [1] | J.J. Stoker, Water Waves, Interscience Publishers, Wiley and Sons, New York (1957). |
| [2] | M.B. Abbott, Computational Hydraulics: Elements of the Theory of Free-Surface Flows, Pitman, London (1979). · Zbl 0406.76002 |
| [3] | J. Cunge, F.M. Holley and A. Verwey, Practical Aspects of Computational River Hydraulics, Pitman, London (1980). |
| [4] | R.J. Fennema and M.H. Chaudry, Simulation of one-dimensional dam-break flows, J. Hyd. Res. 25 (1987) 41-51. · doi:10.1080/00221688709499287 |
| [5] | P. Glaister, An approximate linearised Riemann solver for the one-dimensional Euler equations with real gases, J. Comput. Phys. 74 (1988) 382-408. · Zbl 0632.76079 · doi:10.1016/0021-9991(88)90084-8 |
| [6] | J.P. Vila, Simplified Godunov schemes for 2 x 2 systems of conservation laws, SIAM J. Numer. Anal. 23 (1986) 1173. · Zbl 0623.65091 · doi:10.1137/0723079 |
| [7] | J.P. Vila, Schemas numeriques en hydraulique des écoulements avec discontinuities, Proc. XII Congress IAHR, Lausanne, ed. J.A. Cunge, P. Ackers (1987). |
| [8] | S.K. Godunov, A difference method for the numerical computation of continuous solutions of hydrodynamic equations, Mat. Sbornik, 47 (1959) 271-306; translated as JPRS 7225 by U.S. Dept. of Commerce (1960). · Zbl 0171.46204 |
| [9] | N.N. Yanenko, The Method of Fractional Steps, Springer-Verlag, Berlin (1971). · Zbl 0209.47103 |
| [10] | P. Glaister, Difference schemes for the shallow water equations, Numerical Analysis Report, University of Reading (1987). · Zbl 0967.76068 |
| [11] | P.K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal. 21 (1984) 995. · Zbl 0565.65048 · doi:10.1137/0721062 |
| [12] | P.K. Sweby, A modification of Roe’s scheme for entropy satisfying solutions of scalar non-linear conservation laws, Numerical Analysis Report, University of Reading (1982). |
| [13] | P. Glaister, System of conservation laws with source terms, Numerical Analysis Report, University of Reading (1987). |
| [14] | P. Glaister, Flux difference splitting for the Euler equations with axial symmetry, J. Eng. Math. 22 (1988) 107-121. · Zbl 0663.76070 · doi:10.1007/BF02383596 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
