High resolution finite-element analysis of shallow water equations in two dimensions

Abstract

We present in this study the Taylor–Galerkin finite-element model to stimulate shallow water equations for bore wave propagation in a domain of two dimensions. To provide the necessary precision for the prediction of a sharply varying solution profile, the generalized Taylor–Galerkin finite-element model is constructed through introduction of four parameters. This paper also presents the fundamental theory behind the choice of free parameters. One set of parameters is theoretically determined to obtain the high-order accurate Taylor– Galerkin finite-element model. The other set of free parameters is determined using the underlying discrete maximum principle to obtain the low-order monotonic Taylor–Galerkin finite-element model. Theoretical study reveals that the higher-order scheme exhibits dispersive errors near the discontinuity while lower-order scheme dissipates the discontinuity. A scheme which has a high-resolution shock-capturing ability as a built-in feature is, thus, needed in the present study. Notice that lumping of the mass matrix equations is invoked in the low-order scheme to allow simulation of the hydraulic problem with discontinuities. We check the prediction accuracy against suitable test problems, preferably ones for which exact solutions are available. Based on numerical results, it is concluded that the Taylor–Galerkin-flux-corrected transport (TG-FCT) finite-element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles.