Discrete tangent and adjoint sensitivity analysis for discontinuous solutions of hyperbolic conservation laws

Hüser, Jonathan Jakob; Naumann, Uwe (Thesis advisor); Herty, Michael (Thesis advisor)

Aachen : RWTH Aachen University (2022)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2022


We consider the discrete tangent and adjoint sensitivities computed via algorithmic differentiation of shock capturing numerical methods for hyperbolic conservation laws which are widely used for models of fluid dynamics such as those based on the Euler equations. For discontinuous solutions the discrete sensitivities do not generally converge to the correct sensitivities of the analytical solution as the discretization grid is refined because the analytical sensitivities are singular at the discontinuities of the solution. In this thesis we propose a convergent numerical approximation of the correct sensitivities of shock discontinuities in discontinuous solutions of hyperbolic conservations laws with respect to the parameters of the initial data. We compute the shock sensitivities by approximating the Rankine-Hugoniot condition taking into consideration the numerical viscosity of shock capturing numerical methods in a way that can be computed by algorithmic differentiation tools. The resulting discrete sensitivities enable for example the gradient-based parameter optimization of optimization problems constrained by a hyperbolic conservation law.