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A modified sensitivity equation method for the Euler equations in presence of shocks
Author(s) -
Fiorini Camilla,
Chalons Christophe,
Duvigneau Régis
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22454
Subject(s) - mathematics , sensitivity (control systems) , numerical diffusion , partial differential equation , euler equations , riemann solver , classification of discontinuities , numerical analysis , solver , dirac delta function , backward euler method , convergence (economics) , mathematical analysis , finite volume method , mathematical optimization , physics , electronic engineering , mechanics , engineering , economics , economic growth
The continuous sensitivity equation method allows to quantify how changes in the input of a partial differential equation (PDE) model affect the outputs, by solving additional PDEs obtained by differentiating the model. However, this method cannot be used directly in the framework of hyperbolic PDE systems with discontinuous solution, because it yields Dirac delta functions in the sensitivity solution at the location of state discontinuities. This difficulty is well known from theoretical viewpoint, but only a few works can be found in the literature regarding the possible numerical treatment. Therefore, we investigate in this study how classical numerical schemes for compressible Euler equations can be modified to account for shocks when computing the sensitivity solution. In particular, we propose the introduction of a source term, that allows to remove the spikes associated to the Dirac delta functions in the numerical solution. Numerical studies exhibit a strong impact of the numerical diffusion on the accuracy of this strategy. Therefore, we propose an anti‐diffusive numerical scheme coupled with the approximate Riemann solver of Roe for the state problem. For the sensitivity problem, two different numerical schemes are implemented and compared: one which takes into account the contact wave and another that neglects it. The effects of the numerical diffusion on the convergence of the schemes with respect to the grid are discussed. Finally, an application to uncertainty propagation is investigated and the different numerical schemes are compared.