Optimal design of vertical slot fishways by using shallow water equations

In this paper, we present a mathematical formulation of an optimal design problem related to a vertical slot fishway. The work involves modeling, mathematical analysis and numerical approximation of a coupled problem between a primal hyperbolic system and adjoint problem of shallow water for the cost function of the optimal structure. We express the shape gradient of the cost function by introducing the associated adjoint state system. We proceed with the study of the adjoint system by using the Lax symbolic symmetrizer for hyperbolic systems and pseudo-differential techniques. The numerical resolution of this problem combines two main approaches: The first one relies on the finite volume method with the Roe solver for the spatial and temporal discretization, and the second one uses a minimizing algorithm, the gradient of the objective function, evaluated by an adjoint problem. Numerical simulations are given which illustrate the accuracy of this technique.