Higher-order schemes for the shallow water sensitivity equations with transport
Résumé
The one-dimensional shallow water equations with scalar transport augmented with their direct sensitivity equations form a 6×6 linearly degenerate system. As a consequence, solving the sensitivity equations numerically using shock-capturing schemes requires a specific treatment of discontinuous solutions. The Riemann solvers used for flux calculations must be modified so as to include a coupled discretization of the flux and geometric source terms. The reconstruction in MUSCL-type schemes must also be limited in a more strict way near flow discontinuities. In addition to the second-order, MUSCL-Hancock time stepping method, the first-order MUSCL-HLLG and MUSCL-EVR time stepping algorithms are tested against analytical solutions of the flow and sensitivity equations (asymmetrical rarefaction wave and dambreak problems). Convergence analyses indicate that the second-order time stepping approach is less accurate than first-order ones in a number of cases, while more computationally demanding. The MUSCL-EVR approach provides optimal compromise between solution accuracy and computational efficiency.
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