Abstract
In recent years there has been considerable research into the use of adjoint flow equations for design optimisation (e.g. [Jam95]) and error analysis (e.g. [PGOO, BROI]). In almost every case, the adjoint equations have been formulated under the assumption that the original nonlinear flow solution is smooth. Since most applications have been for incompressible or subsonic flow, this has been valid, however there is now increasing use of such techniques in transonic design applications for which there are shocks. It is therefore of interest to investigate the formulation and discretisation of adjoint equations when in the presence of shocks.
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References
F. Bouchut and F. James. One-dimensional transport equations with discontinuous coefficients. Nonlinear Analysis, 32:891–933, 1998.
R. Becker and R. Rannacher. An optimal control approach to error control and mesh adaptation. In A. Iserles, editor, Acta Numerica 2001. Cambridge University Press, 2001.
M.B. Giles. Adjoint equations and discrete approximations in the presence of shocks. Technical Report NA02/10, Oxford University Computing Laboratory, 2002.
M.B. Giles and N.A. Pierce. On the properties of solutions of the adjoint Euler equations. In M. Baines, editor, Numerical Methods for Fluid Dynamics VI. ICFD, Jun 1998.
M.B. Giles and N.A. Pierce. Analytic adjoint solutions for the quasi-one-dimensional Euler equations. J. Fluid Mech., 426:327–345, 2001.
K.C. Hall, W.S. Clark, and C.B. Lorence. A linearized Euler analysis of unsteady transonic flows in turbomachinery. J. Turbomachinery, 116:477–488, 1994.
A. Jameson. Optimum aerodynamic design using control theory. In M. Hafez and K. Oshima, editors, Computational Fluid Dynamics Review 1995, pages 495–528. John Wiley & Sons, 1995.
D.R. Lindquist and M.B. Giles. Validity of linearized unsteady Euler equations with shock capturing. AIAA J., 32(1):46, 1994.
N.A. Pierce and M.B. Giles. Adjoint recovery of superconvergent functionals from PDE approximations. SIAM Rev., 42(2):247–264, 2000.
E. Tadmor. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal., 28:891–906, 1991.
S. Ulbrich. Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Systems & Control Letters, to appear, 2002.
S. Ulbrich. A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control and Optim., to appear, 2002.
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Giles, M.B. (2003). Discrete Adjoint Approximations with Shocks. In: Hou, T.Y., Tadmor, E. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55711-8_16
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DOI: https://doi.org/10.1007/978-3-642-55711-8_16
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