Adjoint recovery of superconvergent functionals from PDE approximations

Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasi-one-dimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multi-dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

Analysis of Adjoint Error Correction for Superconvergent Functional Estimates

Earlier work introduced the notion of adjoint error correction for obtaining superconvergent estimates of functional outputs from approximate PDE solutions. This idea is based on a posteriori error analysis suggesting that the leading order error term in the functional estimate can be removed by using an adjoint PDE solution to reveal the sensitivity of the functional to the residual error in the original PDE solution. The present work provides a priori error analysis that correctly predicts the behaviour of the remaining leading order error term. Furthermore, the discussion is extended from the case of homogeneous boundary conditions and bulk functionals, to encompass the possibilities of inhomogeneous boundary conditions and boundary functionals. Numerical illustrations are provided for both linear and nonlinear problems.\ud \ud This research was supported by EPSRC under grant GR/K91149, and by NASA/Ames Cooperative Agreement No. NCC 2-5431

An introduction to the adjoint approach to design

Optimal design methods involving the solution of an adjoint system of equations are an active area of research in computational fluid dynamics, particularly for aeronautical applications. This paper presents an introduction to the subject, emphasising the simplicity of the ideas when viewed in the context of linear algebra. Detailed discussions also include the extension to p.d.e.'s, the construction of the adjoint p.d.e. and its boundary conditions, and the physical significance of the adjoint solution. The paper concludes with examples of the use of adjoint methods for optimising the design of business jets.\ud \ud This research was supported by funding from Rolls-Royce plc, BAe Systems plc and EPSRC grants GR/K91149 and GR/L95700

Analytic Adjoint Solutions for the Quasi-1D Euler Equations

The analytic properties of adjoint solutions are examined for the quasi-1D Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green's function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging-diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock.\ud \ud This research was supported by EPSRC under grant GR/K9114

The optical counterpart of SAX J1808.4-3658, the transient bursting millisecond X-ray pulsar

A set of CCD images have been obtained during the decline of the X-ray transient SAX J1808.4-3658 during April-June 1998. The optical counterpart has been confirmed by several pieces of evidence. The optical flux shows a modulation on several nights which is consistent with the established X-ray binary orbit period of 2 hours. This optical variability is roughly in antiphase with the weak X-ray modulation. The source mean magnitude of V=16.7 on April 18 declined rapidly after April 22. From May 2 onwards the magnitude was more constant at around V=18.45 but by June 27 was below our sensitivity limit. The optical decline precedes the rapid second phase of the X-ray decrease by 3 +/- 1 days. The source has been identified on a 1974 UK Schmidt plate at an estimated magnitude of ~20. The nature of the optical companion is discussed.Comment: 5 pages, 3 figures; published in MNRAS, March 15th 199

Accretion column eclipses in the X-ray pulsars GX 1+4 and RX J0812.4-3114

Sharp dips observed in the pulse profiles of three X-ray pulsars (GX 1+4, RX J0812.4-3114 and A 0535+26) have previously been suggested to arise from partial eclipses of the emission region by the accretion column occurring once each rotation period. We present pulse-phase spectroscopy from Rossi X-ray Timing Explorer satellite observations of GX 1+4 and RX J0812.4-3114 which for the first time confirms this interpretation. The dip phase corresponds to the closest approach of the column axis to the line of sight, and the additional optical depth for photons escaping from the column in this direction gives rise to both the decrease in flux and increase in the fitted optical depth measured at this phase. Analysis of the arrival time of individual dips in GX~1+4 provides the first measurement of azimuthal wandering of a neutron star accretion column. The column longitude varies stochastically with standard deviation 2-6 degrees depending on the source luminosity. Measurements of the phase width of the dip both from mean pulse profiles and individual eclipses demonstrates that the dip width is proportional to the flux. The variation is consistent with that expected if the azimuthal extent of the accretion column depends only upon the Keplerian velocity at the inner disc radius, which varies as a consequence of the accretion rate Mdot.Comment: 7 pages, 5 figures, accepted by MNRAS. Included reference

Spectral variation in the X-ray pulsar GX 1+4 during a low-flux episode

The X-ray pulsar GX 1+4 was observed with the RXTE satellite for a total of 51ks between 1996 July 19 - 21. During this period the flux decreased smoothly from an initial mean level of ~ 6 X 10^36 erg/s to a minimum of ~ 4 X 10^35 erg/s (2-60 keV, assuming a source distance of 10 kpc) before partially recovering towards the initial level at the end of the observation. BATSE pulse timing measurements indicate that a torque reversal took place approximately 10 d after this observation. Both the mean pulse profile and the photon spectrum varied significantly. The observed variation in the source may provide important clues as to the mechanism of torque reversals. The single best-fitting spectral model was based on a component originating from thermal photons with kT ~ 1 keV Comptonised by a plasma of temperature kT \~ 7 keV. Both the flux modulation with phase during the brightest interval and the evolution of the mean spectra over the course of the observation are consistent with variations in this model component; with, in addition, a doubling of the column density nH contributing to the mean spectral change. A strong flare of duration 50 s was observed during the interval of minimum flux, with the peak flux ~ 20 times the mean level. Although beaming effects are likely to mask the true variation in Mdot thought to give rise to the flare, the timing of a modest increase in flux prior to the flare is consistent with dual episodes of accretion resulting from successive orbits of a locally dense patch of matter in the accretion disc.Comment: 8 pages, 3 figures, submitted to MNRA