A Method to Calculate Numerical Errors Using Adjoint Error Estimation for Linear Advection
Authors: 
Jeffrey M. Connors, Jeffrey W. Banks, Jeffrey A. Hittinger, Carol S. WoodwardAuthors Info & Claims
Pages 894 - 926
Abstract
This paper is concerned with the computation of numerical discretization error for uncertainty quantification. An a posteriori error formula is described for a functional measurement of the solution to a scalar advection equation that is estimated by finite volume approximations. An exact error formula and computable error estimate are derived based on an abstractly defined approximation of the adjoint solution. The adjoint problem is divorced from the finite volume method used to approximate the forward solution variables and may be approximated using a low-order finite volume method. The accuracy of the computable error estimate provably satisfies an a priori error bound for sufficiently smooth solutions of the forward and adjoint problems. Computational examples are provided that show support of the theory for smooth solutions. The application to problems with discontinuities is also investigated computationally.
References
[1]
M. Ainsworth and J. T. Oden, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Math., 142 (1997), pp. 1--88.
[2]
K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., John Wiley and Sons, New York, 1988.
[3]
G. E. Barter, Shock Capturing with PDE-Based Artificial Viscosity for an Adaptive, Higher-Order Discontinuous Galerkin Finite Element Method, Ph.D. thesis, MIT, Cambridge, MA, 2008.
[4]
T. J. Barth, A posteriori error estimation and mesh adaptivity for finite volume and finite element methods, in Adaptive Mesh Refinement: Theory and Applications, Tomasz Plewa, Timur Linde, and V. Gregory Weirs, eds., Lecture Notes in Computational Sci. Engrg. 41, Springer, Berlin, 2005, pp. 183--202.
[5]
T. J. Barth, Space-time error representation and estimation in Navier-Stokes calculations, in Complex Effects in Large Eddy Simulations, S. C. Kassinos, C. A. Langer, G. Iaccarino, and P. Moin, eds., Lecture Notes in Comput. Sci. Engrg. 56, Springer, Berlin, 2007, pp. 29--48.
[6]
T. J. Barth and M. G. Larson, A posteriori error estimates for higher order Godunov finite volume methods on unstructured meshes, in Finite Volumes for Complex Applications III, R. Herbin and D. Kroner, HERMES Science Publishing, London, 2003.
[7]
O. Benedix and B. Vexler, A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl., 44 (2009), pp. 3--25.
[8]
R. E. Bensow and M. G. Larson, Residual based VMS subgrid modeling for vortex flows, Comput. Methods Appl. Math., 199 (2010), pp. 802--809.
[9]
B. Cockburn, S.-Y. Lin, and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems, J. Comput. Phys., 84 (1989), pp. 90--113.
[10]
B. Cockburn and C.-W. Shu, The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems, J. Comput. Phys., 149 (1998), pp. 199--224.
[11]
C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Math. 1697, Springer, Berlin, 1998, pp. 325--432.
[12]
P. Colella and P. R. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations, J. Comput. Phys., 54 (1984), pp. 174--201.
[13]
D. Estep, M. G. Larson, and R. D. Williams, Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations, Mem. Amer. Math. Soc. 146, AMS, Providence, RI, 2000.
[14]
D. Estep, M. Pernice, D. Pham, S. Tavener, and H. Wang, A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems, J. Comput. Appl. Math, 233 (2009), pp. 459--472.
[15]
D. Estep, M. Pernice, S. Tavener, and H. Wang, A posteriori error analysis for a cut cell finite volume method, Comput. Methods Appl. Math., 200 (2011), pp. 2768--2781.
[16]
D. Estep, S. Tavener, and T. Wildey, A posteriori error estimation and adaptive mesh refinement for a multiscale operator decomposition approach to fluid-solid heat transfer, J. Comput. Phys., 229 (2010), pp. 4143--4158.
[17]
K. J. Fidkowski and D. L. Darmofal, A triangular cut-cell adaptive method for high-order discretizations of the compressible Navier-Stokes equations, J. Comput. Phys., 225 (2007), pp. 1653--1672.
[18]
K. J. Fidkowski and D. L. Darmofal, Output-based error estimation and mesh adaptation in computational fluid dynamics: Overview and recent results, in Proceedings of the 47th AIAA Aerospace Sciences Meeting, 2009.
[19]
K. J. Fidkowski and P. L. Roe, An entropy adjoint approach to mesh refinement, SIAM J. Sci. Comput., 32 (2010), pp. 1261--1287.
[20]
J. E. Fromm, A method for reducing dispersion in convective difference schemes, J. Comput. Phys., 3 (1968), pp. 176--189.
[21]
M. B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11 (2002), pp. 145--236.
[22]
A. Harten, B. Engquist, S. Osher, and S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes, III, J. Comput. Phys., 71 (1987), pp. 231--303.
[23]
A. Harten, S. Osher, B. Engquist, and S. R. Chakravarthy, Some results on uniformly high-order accurate essentially nonoscillatory schemes, Appl. Numer. Math., 2 (1986), pp. 347--377.
[24]
R. Hartmann, Multitarget error estimation and adaptivity in aerodynamic flow simulations, SIAM J. Sci. Comput., 31 (2008), pp. 708--731.
[25]
R. Hartmann and P. Houston, Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws, SIAM J. Sci. Comput., 24 (2002), pp. 979--1004.
[26]
R. Hartmann and P. Houston, Adaptive discontinuous Galerkin finite element methods for the compressible Euler equations, J. Comput. Phys., 183 (2002), pp. 508--532.
[27]
G. I. Marchuk, V. I. Agoshkov, and V. P. Shutyaev, Adjoint Equations and Perturbation Algorithms in Nonlinear Problems, CRC Press, Boca Raton, FL, 1996.
[28]
W. L Oberkampf, T. G Trucano, and C. Hirsch, Verification, validation, and predictive capability in computational engineering and physics, Appl. Mech. Rev., 57 (2004), pp. 345--384.
[29]
N. A. Pierce and M. B. Giles, Adjoint and defect error bounding and correction for functional estimates, J. Comput. Phys., 200 (2004), pp. 769--794.
[30]
P. J. Roache, Verification and Validation in Computational Science and Engineering, Hermosa Publishers, Socorro, NM, 1998.
[31]
C. J. Roy, Review of code and solution verification procedures for computational simulation, J. Comput. Phys., 205 (2005), pp. 131--156.
[32]
C. Steiner and S. Noelle, On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control, Int. J. Numer. Methods Biomed. Engrg., 26 (2010), pp. 790--806.
[33]
B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys., 32 (1979), pp. 101--136.
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© 2013, Society for Industrial and Applied Mathematics.
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Published: 01 January 2013
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