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 stabi lity 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 param eters is determined from the underlying discrete maximum principle to obtai n the monotonic solutions. The resulting two models are used in combination through the flux correct transport technique of Zalesak, thereby construct ing a finite element model which has the ability to capture hydraulic disco ntinuities. In addition, this paper highlights the implementation of two Kr ylov subspace iterative solvers, namely, the bi-conjugate gradient stabiliz ed (Bi-CGSTAB) and the generalized minimum residual (GMRES) methods. For th e sake of comparison, the multifrontal direct solver is also considered. Th e performance characteristics of the investigated solvers are assessed usin g results of a standard test widely used as a benchmark in hydraulic modeli ng. Based on numerical results, it is shown that the present finite element method can render the technique suitable for solving shallow water equatio ns with sharply varying solution pro les. Also, the GMRES solver is shown t o have a much better convergence rate than the Bi-CGSTAB solver, thereby sa ving much computing time compared to the multifrontal solver.