Two Element-by-Element Iterative Solutions for Shallow Water Equations

In this paper we apply the generalized Taylor–Galerkin finite element model to simulate bore wave propagation in a domain of two dimensions. For stability and accuracy reasons, we generalize the model through the introduction of four free parameters. One set of parameters is rigorously determined to obtain the high-order finite element solution. The other set of free parameters is determined from the underlying discrete maximum principle to obtain the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby constructing a finite element model which has the ability to capture hydraulic discontinuities. In addition, this paper highlights the implementation of two Krylov subspace iterative solvers, namely, the bi-conjugate gradient stabilized (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For the sake of comparison, the multifrontal direct solver is also considered. The performance characteristics of the investigated solvers are assessed using results of a standard test widely used as a benchmark in hydraulic modeling. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles. Also, the GMRES solver is shown to have a much better convergence rate than the Bi-CGSTAB solver, thereby saving much computing time compared to the multifrontal solver.