Sensitivity analysis for nonlinear hyperbolic systems of conservation laws

Abstract

Sensitivity analysis (SA) concerns the quantification of changes in Partial Differential Equations (PDEs) solution due to perturbations in the model input. Standard SA techniques for PDEs, such as the continuous sensitivity equation method, rely on the differentiation of the state variable. However, if the governing equations are hyperbolic PDEs, the state can exhibit discontinuities yielding Dirac delta functions in the sensitivity. We aim at modifying the sensitivity equations to obtain a solution without delta functions. This is motivated by several reasons: firstly, a Dirac delta function cannot be seized numerically, leading to an incorrect solution for the sensitivity in the neighbourhood of the state discontinuity; secondly, the spikes appearing in the numerical solution of the original sensitivity equations make such sensitivities unusable for some applications. Therefore, we add a correction term to the sensitivity equations. We do this for a hierarchy of models of increasing complexity: starting from the inviscid Burgers’ equation, to the quasi 1D Euler system. We show the influence of such correction term on an optimization algorithm and on an uncertainty quantification problem.