| Summary: | 博士 === 國立臺灣大學 === 土木工程學研究所 === 92 === The development of a numerical scheme that resolves sharp discontinuities without spurious oscillations and do not produce too much numerical dissipation is of great importance in the computational shallow-water hydrodynamics. In this thesis, three hybrid flux-splitting finite-volume schemes are proposed for solving two-dimensional shallow water equations.
In the framework of the finite volume method, a hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method (AUSM) to estimate the cell-interface fluxes. Based on the proposed algorithm, a first-order hybrid flux-splitting finite-volume (HFS) scheme is developed, which is robust and rather simple to implement. To improve the numerical resolutions of discontinuities, the monotonic upstream schemes for conservation laws (MUSCL) method with limiters and the two-step component-wise total variation diminishing (TVD) method are adopted for the second-order extensions.
The proposed three finite-volume schemes are verified through the simulations of the 1D idealized dam-break, extreme rarefaction wave, steady transcritical flow and oblique hydraulic jump problems. The numerical results by the proposed schemes are compared with those by other shock-capturing upwind schemes as well as exact solutions. It is demonstrated that the proposed schemes are accurate and efficient to capture the discontinuous solutions without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. In addition, the proposed schemes are proven to produce no entropy-violating solution and to achieve the benefits combining the efficiency of flux-vector splitting (FVS) scheme and the accuracy of flux-difference splitting (FDS) scheme.
Furthermore, the proposed schemes are applied to simulate several 2D dam-break problems, including the partial dam breaking, circular dam breaking and four experimental dam-break problems. The simulated results show that the proposed schemes can deal with the rarefaction waves, shocks, the reflected shocks, the reverse flows and the dry/wet fronts very well.
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