Adaptive finite element methods for solving Saint-Venant equations

Solving Saint-Venant equations by the finite element method needs long CPU time (even for a short time). Moreover, if the channel length is fairly large, the system resulted by discretization is not directly solvable, and one should use iterative methods. Therefore, total error is error resulted by discretization and by solving the system by iterative methods. In this paper, we apply three adaptive finite element methods for solving Saint-Venant equations to obtain, first, a better approximated solution and, second, a significant decrease in CPU time and computational complexity. In the first method, a coarse grid is considered and, by partitioning a few intervals, the problem is solved on this new coarse grid. In the second method, the problem is solved for the first few moments and, then, using the regression method, the solution is obtained in the following time. In the third method, like the second method, the problem is solved for the first few moments and, then, the approximated solution is predicted for following times. Finally, two numerical examples for supercritical and subcritical flows are given to support our results.