Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: Linearized approximations and linearized output functionals

SIAM Journal on Numerical Analysis

Volumn 48, Issue 3, 2010, Pages 882-904

  Giles, Mike ^a   Ulbrich, Stefan ^b  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Adjoint; Conservation law; Hyperbolic; Linearized; Numerical analysis

Indexed keywords

ADJOINTS; ASYMPTOTIC BEHAVIORS; CONSERVATION LAW; DISCONTINUOUS SOLUTIONS; DISCRETE APPROXIMATION; DISCRETE EQUATIONS; DISCRETIZATIONS; FLUX FUNCTION; FUNCTIONALS; HYPERBOLIC; HYPERBOLIC EQUATIONS; KEYPOINTS; LINEARIZED; LINEARIZED EQUATIONS; NUMERICAL RESULTS; POINTWISE CONVERGENCE; UNIFORM GRIDS;

ASYMPTOTIC ANALYSIS; CONVERGENCE OF NUMERICAL METHODS; HYPERBOLIC FUNCTIONS; NUMERICAL ANALYSIS;

LINEARIZATION;

EID: 77956028827     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/080727464     Document Type: Article

References (29)

1. W. K. ANDERSON AND V. VENKATAKRISHNAN, Aerodynamic design optimization on unstructured grids with a continuous adjoint formulation, Comput. & Fluids, 28 (1999), pp. 443-480.
2. T. BARTH AND H. DECONINCK, EDS., Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Lect. Notes Comput. Sci. Eng. 25, Springer-Verlag, Berlin, 2003.
3. F. BOUCHUT AND F. JAMES, One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal., 32 (1998), pp. 891-933.
4. C. M. BENDER AND S. A. ORSZAG, Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill, New York, 1978.
5. C. BARDOS AND O. PIRONNEAU, Derivatives and control in the presence of shocks, Comput. Fluid Dynam. J., 11 (2003), pp. 383-392.
6. R. BECKER AND R. RANNACHER, An optimal control approach to error control and mesh adaptation, in Acta Numerica 2001, A. Iserles, ed., Cambridge University Press, Cambridge, UK, 2001, pp. 1-102.
7. M. CRANDALL AND A. MAJDA, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), pp. 1-21.
8. P. COURTIER AND O. TALAGRAND, Variational assimilation of meteorological observations with the adjoint vorticity equation, II, Numerical results, Q. J. R. Meteorol. Soc., 113 (1987), pp. 1329-1347.
9. G. DAL MASO, P. G. LEFLOCH, AND F. MURAT, Definition and weak stability of non - conservative products, J. Math. Pures Appl. (9), 74 (1995), pp. 483-548.
10. M. B. GILES, Analysis of the accuracy of shock-capturing in the steady quasi-1D Euler equations, Internat. J. Comput. Fluid Dynam., 5 (1996), pp. 247-258.
11. M. B. GILES, Discrete adjoint approximations with shocks, in Hyperbolic Problems: Theory, Numerics, Applications, T. Hou and E. Tadmor, eds., Springer-Verlag, New York, 2003.
12. M. B. GILES AND N. A. PIERCE, On the properties of solutions of the adjoint Euler equations, in Numerical Methods for Fluid Dynamics VI, M. Baines, ed., ICFD, Reading, UK, 1998.
13. M. B. GILES AND N. A. PIERCE, Analytic adjoint solutions for the quasi-one-dimensional Euler equations, J. Fluid Mech., 426 (2001), pp. 327-345.
14. M. GILES AND S. ULBRICH, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: Adjoint approximations and extensions, SIAM J. Numer. Anal., 48 (2010), pp. 905-921.
15. K. C. HALL, W. S. CLARK, AND C. B. LORENCE, A linearized Euler analysis of unsteady transonic flows in turbomachinery, J. Turbomachinery, 116 (1994), pp. 477-488.
16. T. Y. Hou AND P. G. LEFLOCH, Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp., 62 (1994), pp. 497-530.
17. C. HOMESCU AND I. M. NAVON, Optimal control of flow with discontinuities, J. Comput. Phys., 187 (2003), pp. 660-682.
18. S. I. HARIHARAN AND G. W. YOUNG, Comparison of asymptotic solutions of a phase-field model to a sharp-interface model, SIAM J. Appl. Math., 62 (2001), pp. 244-263.
19. A. JAMESON, Optimum aerodynamic design using control theory, in Computational Fluid Dynamics Review 1995, M. Hafez and K. Oshima, eds., John Wiley & Sons, New York, 1995, pp. 495-528.
20. J. KEVORKIAN AND J. D. COLE, Perturbation Methods in Applied Mathematics, Appl. Math. Sci. 34, Springer-Verlag, New York, 1981.
21. P. LEFLOCH, An existence and uniqueness result for two nonstrictly hyperbolic systems, in Nonlinear Evolution Equations That Change Type, IMA Vol. Math. Appl. 27, Springer-Verlag, New York, 1990, pp. 126-138.
22. D. R. LINDQUIST AND M. B. GILES, Validity of linearized unsteady Euler equations with shock capturing, AIAA J., 32 (1994), pp. 46-53.
23. H. NESSYAHU AND E. TADMOR, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal., 29 (1992), pp. 1505-1519.
24. J. STOER AND R. BULIRSCH, Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
25. K. SREENIVAS AND D. L. WHITFIELD, Time- and frequency-domain numerical simulation of linearized Euler equations, AIAA J., 36 (1998), pp. 968-975.
26. E. TADMOR, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 28 (1991), pp. 891-906.
27. O. TALAGRAND AND P. COURTIER, Variational assimilation of meteorological observations with the adjoint vorticity equation, I, Theory, Q. J. R. Meteorol. Soc, 113 (1987), pp. 1311-1328.
28. S. ULBRICH, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim., 41 (2002), pp. 740-797.
29. S. ULBRICH, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws, Systems Control Lett., 48 (2003), pp. 313-328.