采用和谐的加权平均通量(WAF)算法，研究了浅水波方程的间断解及污染物传输问题。该算法采用WAF格式和HLLCRiemann求解器近似单元边界数值通量，中心差分格式离散地形源项，然后理论上证明了该算法是和谐的。最后利用WAF算法对非平底地形上浅水波间断解及污染物传输问题进行数值计算，精确地捕捉到了间断解和污染物运动过程，结果表明该算法满足守恒性，具有高分辨率、无振荡及捕捉污染物运动边界的能力。
A well-balanced weighted average flux (WAF) scheme is proposed for the nonhomogeneous shallow water equations with pollutant transport. The water surface elevation and water discharge are used as the conserved variables in the shallow water equations, while the conserved variables and the bed elevation are set on the staggered mesh. Here, the WAF scheme combined with the HLLC Riemann solver is directly performed for the nonhomogeneous shallow water equations with pollutant transport and the spatial second order central difference approximation is applied to source terms. Then, it’s proved that the extended WAF scheme satisfies the equilibrium property in theory. Finally, several numerical tests about the steady and nonsteady flows over the irregular bottom or with the wet/dry front are simulated. The results obviously suggest that the WAF scheme is a well-balanced, robust, efficient, practical and high resolution scheme.
[1] GODUNOV S K. Finite Difference Methods for the Computation of Discontinuous Solutions of the Equations of Fluid Dynamics[J]. Matematicheskiǐ Sbornik, 1959, 47: 271-306.
[2] TORO E F. A Weighted Average Flux Method for Hyperbolic Conservation Laws[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1989, 423(1865): 401-418.
[3] ROE L P. Upwind Differencing Schemes for Hyperbolic Conservation Laws with Source Terms [C]∥Springer-Verlag. Proceedings of an Advanced Research Workshop: Nonlinear Hyperbolic Problems: Lecture Notes in Mathematics, Volume 1270. St. Etienne, France January 13-17, 1986: 41-51.
[4]BERMUDEZ A, VAZQUEZ-CENDON E M. Upwind Methods for Hyperbolic Conservation Laws with Source Terms [J]. Computers and Fluids,1994,23(8):1049-1071.
[5] HUBBARD E M, GARCIA-NAVARRO P. Flux Difference Splitting and the Balancing of Source Terms and Flux Gradients[J]. Journal of Computational Physics, 2001, 165(1): 89-125.
[6] ZHOU G J, CAUSON M D, MINGHAN G C, et al. The Surface Gradient Method for the Treatment of Source Terms in the Shallow-Water Equations [J]. Journal of Computational Physics, 2001, 168(1): 1-25.
[7] GALLOUT T, HRARD J M, SEGUIN N. Some Approximate Godunov Schemes to Compute Shallow Water Equations with Topography[J]. Computers and Fluids, 2003, 32(4): 479-513.
[8]TORO E F. Shock-Capturing Methods for Free-Surface Shallow Flows[M]. Chichester: Wliey, 2001.
[9]FERNANDEZ-NIETO E D, NARBONA-REINA G. Extension of WAF Type Methods to Non-homogeneous Shallow Water Equations with Pollutant[J]. Journal of Scientific Computing, 2008, 36(2): 193-217.
