Discontinuous Galerkin Method for 1D Shallow Water Flow in Nonrectangular and Nonprismatic Channels

Abstract

A total variation diminishing Runge-Kutta discontinuous Galerkin finite element method for the solution of one-dimensional (1D) shallow water flow equations for natural channels is presented. The hydrostatic pressure force and the wall pressure force terms are combined to simplify the calculations and prevent unphysical flow attributable to improper treatment of the bottom slope term. The treatment of the combined term that appropriately accounts for the momentum flux is given. HLL and Roe Riemann solvers are assessed for the mass and momentum flux terms. Numerical tests are conducted using prismatic rectangular and nonrectangular channels as well as non prismatic channels and natural channel for dam break, supercritical flow, transcritical flow, and dry-bed problems. Slope limiters based on flow cross section area, water surface, and water depth are evaluated. The tests show that HLL and Roe solvers provide similar accuracy. However, the slope limiter based on flow area provides more accurate solutions for tests in nonrectangular and natural channels.