SUPG formulation augmented with YZβ shock‐capturing for computing shallow‐water equations

Abstract

AbstractWe demonstrate that the streamline‐upwind/Petrov–Galerkin (SUPG) formulation enhanced with YZβ discontinuity‐capturing, that is, the SUPG‐YZβ formulation, is an efficient and robust method for computing 2D shallow‐water equations (SWEs). The SUPG‐stabilized semi‐discrete formulation is discretized in time by employing the backward Euler time‐integration scheme. The nonlinear equation systems arising from the space and time discretizations are handled using the Newton–Raphson (N–R) method at each time step. The resulting linear equation systems are solved directly at each nonlinear iteration. Two challenging test problems are provided to examine the performance of the proposed formulation and techniques. To that end, we consider a full dam‐break and a partial dam‐break problem. We develop the solvers in the FEniCS environment. Test computations reveal that the SUPG‐YZβ formulation successfully eliminates spurious oscillations that cannot be captured with the SUPG‐stabilized formulation alone in narrow regions where steep gradients occur.