Numerical Simulation Of Two-Dimensional Shallow Water Flows

One of the primary trends of the modern shallow-water flows numerical simulation is to make use of the mathematic similarity between the homogeneous shallow water equation and Euler equation, as well the high performance algorithm of computational aerodynamics such as Osher, Roe, FVS, HLL, HLLC, ENO and WENO to simulate flow that contains discontinuity or weak discontinuity such as dam-break and bore. The quotations will be adjusted in accordance with the particularity of shallow-water flows. Based on the research of other scholars, this paper uses the finite volume method with high performance schemes to solve two-dimensional shallow water equations and builds a mathematical model that can stimulate two-dimensional shallow-water flows on unstructured grids (triangular or quadrilateral) .Generally speaking, considerable numerical dissipation will happen with the first-order scheme. Therefore MUSCL scheme is used to build second-order schemes in this paper. The common method of gradient reconfiguration is designed in the light of one-dimensional or two-dimensional, while methods that consider two-dimensional unstructured grids are not familiar. That's because the arbitrariness caused by the positions of nodes and the units around the nodes makes it more difficult to achieve the high order reconfiguration on unstructured grids. And it largely depends on the relationship of units to extend one-dimensional reconfiguration to two-dimensional unstructured grids. The Gauss-Green formula and Linear Least-Squares are adopted to solve the gradient equation. The influences of nearby contiguous units are well considered and gradients reconfigured in two different ways are analyzed to get a more rational numerical solution according to the weighted modification.