| The shallow water equations have a wide range of applications in the ocean, river, climate modeling, marine biology and environmental engineering.Related issues include storm surges, tsunamis and floods prediction, sediment and contaminant transport, etc.A second-order Runge-Kutta discontinuous Galerkin finite element model is built based on one-dimensional shallow water equations, and is applied to numerical simulation of discontinuous shallow flows on irregular bottom topography successfully. The main contents and conclusions are as follows:(1) The model is based on conservation forms of the shallow water equations. The computational domain is partitioned into a set of elements, and then the equations are integrated over the element. For the space discretization in the model, the HLL or HLLC approximate Riemann solver as the numerical flux at interfaces of elements is employed. For the time discretization, a third-order Runge-Kutta scheme which is TVD, is applied.(2) In the model, a slope limiter or stabilization operator only around discontinuities using discontinuity detector is used in order to alleviate spurious oscillations near discontinuities. A discretization of the source terms is applied, so the model can solve the discontinuous shallow water flows with complex bed topography. So a well-balanced discontinuous Galerkin finite element model is constructed finally.(3) The model is applied to numerical simulation of some classic discontinuous flow examples to prove its validity and applicability. The examples include idealized dam-break problem, a small perturbation of a steady-state water, transcritical flow over a hump, hydraulic jump, tidal waves and tidal bores. When solving discontinuous shallow water flow, the situations of non-prismatic channels and discontinuous bed topography can be considered. All the computations are good agreement with the analytical solutions, and the results exclude non-physical spurious oscillations near the shock. |
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