Upwind finite volume solution of sensitivity equations for hyperbolic systems of conservation laws with discontinuous solutions

An upwind, finite volume method is proposed for the direct solution of one-dimensional, hyperbolic systems-based sensitivity equations. Sensitivity equations for hyperbolic systems of conservation laws require a specific treatment of discontinuities, across which Dirac source terms appear, thus leading to modify the classical jump relationships. The modified jump relationships are used to derive an extension of the HLL and HLLC Riemann solvers for the solution of one-dimensional, hyperbolic systems-based sensitivity equations. The solver is developed for 3 × 3 systems where the central wave is a contact wave. A specific treatment is needed to preserve the invariance property of the third component of the sensitivity vector along the contact wave. The proposed solver is applied to the one-dimensional, Saint Venant equations with passive, scalar advection and tested successfully against analytical solutions including the influence of topography-induced source terms in both the flow and sensitivity equations.