Discrete Adjoint Approximations with Shocks
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 assump
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1 Introduction 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. The reason that shocks present a problem is that the adjoint equat ions are defined to be adjoint to the equations obtained by linearising the original nonlinear flow equations. Therefore, this raises the whole issue of linearised perturbations to the shock. The validity of linearised shock capturing for harmonically oscillating shocks in flutter analysis was investigated by Lindquist and Giles [LG94] who showed that the shock capturing produces the correct prediction of integral quantities such as unsteady lift and moment provided the shock is smeared over a number of grid points. As a result , linearised shock capturing is now the standard method of turbomachinery aeroelastic analysis [HCL94], benefitting from the computational advant ages of the linearised approach, without the many drawbacks of shock fitting. There has been very little prior research into adjoint equations for flows with shocks . Giles and Pierce [GPOI] have shown that the analytic derivation of the adjoint equations for the steady quasi-one-dimensional Euler equations requires the specification of an internal adjoint boundary condition at the shock. However, the numerical evidence [GP98] is that the correct adjoint solution is obtained using either the "fully discrete" approach (in which one linearises the discrete equations and uses the transpose) or the "continuous" approach (in which one discretises the analytic adjoint equations). It is not clear though that this will remain true in two dimensions, for which there is a similar adjoint boundary condition along a shock. In this paper, we consider unsteady one-dimensional hyperbolic equations with a convex scalar flux, and in particular obtain numerical results for Burgers equation. Tadmor [Tad91] developed a Lip' topology for the formulation of adjoint equations for this problem, with application to linear post-processing functionals. Building on this and the work of Bouchut
T.Y. Hou et al.(eds.), Hyperbolic Problems: Theory, Numerics, Applications © Springer-Verlag Berlin Heidelberg 2003
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M.B. Giles
and James [BJ98], Ulbrich has very recently introduced the concept of shiftdifferentiability [Ulb02a, Ulb02b] to handle nonlinear functionals of the type considered in this paper. This supplies the analytic adjoint solution against which the numerical solutions in this paper will be compared. An alternative derivation of this analytic s
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