×

Analysis of the properties of adjoint equations and accuracy verification of adjoint model based on FVM. (English) Zbl 1470.76062

Summary: There are two different approaches on how to formulate adjoint numerical model (ANM). Aiming at the disputes arising from the construction methods of ANM, the differences between nonlinear shallow water equation and its adjoint equation are analyzed; the hyperbolicity and homogeneity of the adjoint equation are discussed. Then, based on unstructured meshes and finite volume method, a new adjoint model was advanced by getting numerical model of the adjoint equations directly. Using a gradient check, the correctness of the adjoint model was verified. The results of twin experiments to invert the bottom friction coefficient (Manning’s roughness coefficient) indicate that the adjoint model can extract the observation information and produce good quality inversion. The reason of disputes about construction methods of ANM is also discussed in the paper.

MSC:

76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Thacker, W. C.; Long, R. B., Fitting dynamic to data, Journal of Geophysical Research, 93, 1227-1240 (1988)
[2] Le Dimet, F.-X.; Talagrand, O., Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects, Tellus A, 38, 2, 97-110 (1986)
[3] Bennett, A. F., Inverse Methods in Physical Oceanography (1992), New York, NY, USA: Cambridge University Press, New York, NY, USA · Zbl 0782.76002
[4] Heemink, A. W.; Mouthaan, E. E. A.; Roest, M. R. T.; Vollebregt, E. A. H.; Robaczewska, K. B.; Verlaan, M., Inverse 3D shallow water flow modelling of the continental shelf, Continental Shelf Research, 22, 3, 465-484 (2002)
[5] Peng, S.-Q.; Xie, L.; Pietrafesa, L. J., Correcting the errors in the initial conditions and wind stress in storm surge simulation using an adjoint optimal technique, Ocean Modelling, 18, 3-4, 175-193 (2007)
[6] Zhang, A.; Wei, E.; Parker, B. B., Optimal estimation of tidal open boundary conditions using predicted tides and adjoint data assimilation technique, Continental Shelf Research, 23, 11-13, 1055-1070 (2003)
[7] Talagrand, O.; Courtier, P., Variational assimilation of meteorological observations with the adjoint vorticity equation. I: theory, Quarterly Journal of the Royal Meteorological Society, 113, 478, 1311-1328 (1987)
[8] Schroter, J.; Seiler, U.; Wenzel, M., Variational assimilation of geosat data into an eddy-resolving model of the gulf stream extension area, Journal of Physical Oceanography, 23, 5, 925-953 (1993)
[9] Sirkes, Z.; Tziperman, E., Finite difference of adjoint or adjoint of finite difference?, Monthly Weather Review, 125, 12, 3373-3378 (1997)
[10] Lu, X.-Q.; Wu, Z.-K.; Gu, Y.; Tian, J.-W., Study on the adjoint method in data assimilation and the related problems, Applied Mathematics and Mechanics, 25, 6, 636-646 (2004) · Zbl 1091.86503
[11] Chen, C.; Zhu, J.; Ralph, E.; Green, S. A.; Budd Wells, J.; Zhang, F. Y., Prognostic modeling studies of the Keweenaw Current in Lake Superior. Part I: formation and evolution, Journal of Physical Oceanography, 31, 2, 379-395 (2001)
[12] Chen, C.; Zhu, J.; Zheng, L.; Ralph, E.; Budd, J. W., A non-orthogonal primitive equation coastal ocean circulation model: application to Lake Superior, Journal of Great Lakes Research, 30, 1, 41-54 (2004)
[13] Fang, F.; Pain, C. C.; Piggott, M. D.; Gorman, G. J.; Goddard, A. J. H., An adaptive mesh adjoint data assimilation method applied to free surface flows, International Journal for Numerical Methods in Fluids, 47, 8-9, 995-1001 (2005) · Zbl 1134.86004
[14] Fang, F.; Piggott, M. D.; Pain, C. C.; Gorman, G. J.; Goddard, A. J. H., An adaptive mesh adjoint data assimilation method, Ocean Modelling, 15, 1-2, 39-55 (2006)
[15] Lu, C.; Qiu, J.; Wang, R., Weighted essential non-oscillatory schemes for tidal bore on unstructured meshes, International Journal for Numerical Methods in Fluids, 59, 6, 611-630 (2009) · Zbl 1156.76038
[16] Chen, C.; Beardsley, R. C.; Cowles, G.
[17] Roe, P. L., Approximate Riemann solvers, parameter vectors, and difference schemes, Journal of Computational Physics, 43, 2, 357-372 (1981) · Zbl 0474.65066
[18] Ambrosi, D., Approximation of shallow water equations by Roe’s approximate Riemann solver, International Journal for Numerical Methods in Fluids, 20, 157-168 (1995) · Zbl 0831.76063
[19] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (1997), New York, NY, USA: Springer, New York, NY, USA · Zbl 0888.76001
[20] Chen, Y.-D.; Wang, R.-Y.; Li, W.; Lu, C.-N., Inversing open boundary conditions of tidal bore estuary using adjoint data assimilation method, Advances in Water Science, 18, 6, 829-833 (2007)
[21] Navon, I. M.; Zou, X.; Derber, J.; Sela, J., Variational data assimilation with an adiabatic version of the NMC spectral model, Monthly Weather Review, 120, 7, 1433-1446 (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.