Comparison study of some finite volume and finite element methods for the shallow water equations with bottom topography and friction terms
Number in series
Preprint. Published in: Z. angew. Math. Mech. (ZAMM), 86(11), 2006, 874–891
We present a comparison of two discretization methods for the shallow water equations, namely the finite volume method and the finite element scheme. A reliable model for practical interests includes terms modelling the bottom topography as well as the friction effects. The resulting equations belong to the class of systems of hyperbolic partial differential equations of first order with zero order source terms, the so-called balance laws. In order to approximate correctly steady equilibrium states we need to derive a well-balanced approximation of the source term in the finite volume framework. As a result our finite volume method, a genuinely multidimensional finite volume evolution Galerkin (FVEG) scheme, approximates correctly steady states as well as their small perturbations (quasi-steady states). The second discretization scheme, which has been used for practical river flow simulations, is the finite element method (FEM). In contrary to the FVEG scheme, which is a time explicit scheme, the FEM uses an implicite time discretization and the Newton-Raphson iterative scheme for inner iterations. We compare the accuracy and performance of both scheme through several numerical experiments, which demonstrate the reliability of both discretization techniques and correct approximation of quasisteady states with bottom topography and friction.
well-balanced schemes, steady states, systems of hyperbolic balance laws, shallow water equations, evolution Galerkin schemes