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

We present in this study the Taylor-Galerkin finite-element model to stimul ate shallow water equations for bore wave propagation in a domain of two di mensions. To provide the necessary precision for the prediction of a sharpl y varying solution profile, the generalized Taylor-Galerkin finite-element model is constructed through introduction of four parameters. This paper al so presents the fundamental theory behind the choice of free parameters. On e set of parameters is theoretically determined to obtain the high-order ac curate Taylor-Galerkin finite-element model. The other set of free paramete rs is determined using the underlying discrete maximum principle to obtain the low-order monotonic Taylor-Galerkin finite-element model. Theoretical s tudy reveals that the higher-order scheme exhibits dispersive errors near t he discontinuity while lower-order scheme dissipates the discontinuity. A s cheme which has a high-resolution shock-capturing ability as a built-in fea ture 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 accura cy against suitable test problems, preferably ones for which exact solution s are available. Based on numerical results, it is concluded that the Taylo r-Galerkin-flux-corrected transport (TG-FCT) finite-element method can rend er the technique suitable for solving shallow water equations with sharply varying solution profiles. (C) 2001 Elsevier Science B.V. All rights reserv ed.