Simulation of unsteady flow over floodplain using the diffusive wave equation and the modified finite element method

Abstract

► We consider solution of 2D nonlinear diffusive wave equation in a floodplain. ► We modify the Galerkin finite element method with triangular elements. ► For time integration we use a two-stage difference scheme. ► The modified finite element method leads to more effective algorithm of solution. ► The moving shoreline is obtained as a result of solution at fixed grid point. We consider solution of 2D nonlinear diffusive wave equation in a domain temporarily covered by a layer of water. A modified finite element method with triangular elements and linear shape functions is used for spatial discretization. The proposed modification refers to the procedure of spatial integration and leads to a more general algorithm involving a weighting parameter. The standard finite element method and the finite difference method are its particular cases. Time integration is performed using a two-stage difference scheme with another weighting parameter. The resulting systems of nonlinear algebraic equations are solved using the Picard and Newton iterative methods. It is shown that the two weighting parameters determine the accuracy and stability of the numerical solution as well as the convergence of iterative process. Accuracy analysis using the modified equation approach carried out for linear version of the governing equation allowed to evaluate the numerical diffusion and dispersion generated by the method as well as to explain its properties. As the finite element method accounts for the Neumann type of boundary conditions in a natural way, no special treatment of the boundary is needed. Consequently the problem of moving grid point, which must follow the shoreline, in the proposed approach is overcome automatically. The current position of moving boundary is obtained as a result of solution of the governing equation at fixed grid point.