An element-free Galerkin method for simulation of stationary two-dimensional shallow water flows in rivers

Abstract

In this paper, the meshless method is introduced to the hydraulics. An element-free Galerkin (EFG) method for simulation of two-dimensional shallow water flows in rivers is presented, and its implementation is described. In this method only the nodal data which may be the same as those used in the finite element methods (FEMs) and a description of the domain boundary geometry are necessary; no element or grid connectivity is needed. This makes the method particularly attractive for modelling shallow water flows in rivers for which the mesh generation is usually very difficult because of very irregular topography and strongly varied roughness of the river bottoms. In the EFG method the moving least-squares interpolation is used to construct the trial functions. The modelled domain is represented through the nodal points. A Galerkin method is applied to discretise the governing differential equations, resulting in a simultaneous equation system. An underlying cell structure for calculation of the integrals in Galerkin equations is used. The key advantages of the EFG method in comparison with the FEM are that the method is meshless and the independent variables and their gradients are continuous in the entire domain. The EFG method is also advantageous by changing or refining the nodal distribution in the domain. In addition, the Babuška–Brezzi condition is satisfied.