Sensitivity equations for hyperbolic conservation law-based flow models

Abstract

The present paper focuses on the governing equations for the sensitivity of the variables to the parameters in flow models that can be described by one-dimensional scalar, hyperbolic conservation laws. The sensitivity is shown to obey a hyperbolic, scalar conservation law. The sensitivity is a conserved scalar except in the case of discontinuous flow solutions, where an extra, point source term must be added to the equations in order to enforce conservation. The propagation speed of the sensitivity waves being identical to that of the conserved variable in the original conservation law, the system of conservation laws formed by the original hyperbolic equation and the equation satisfied by the sensitivity is linearly degenerate. A consequence on the solution of the Riemann problem is that rarefaction waves for the variable of the original equation result in vacuum regions for the sensitivity. The numerical solution of the hyperbolic conservation law for the sensitivity by finite volume methods requires the implementation of a specific shock detection procedure. A set of necessary conditions is defined for the discretisation of the source term in the sensitivity equation. An application to the one-dimensional kinematic wave equation shows that the proposed numerical technique allows analytical solutions to be reproduced correctly. The computational examples show that first-order numerical schemes do not yield satisfactory numerical solutions in the neighbourhood of moving shocks and that higher-order schemes, such as the MUSCL scheme, should be used for sharp transients.