Calcolo

, Volume 46, Issue 3, pp 157–185

Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation

  • Ngoc-Cuong Nguyen
  • Gianluigi Rozza
  • Anthony T. Patera
Article

DOI: 10.1007/s10092-009-0005-x

Cite this article as:
Nguyen, NC., Rozza, G. & Patera, A.T. Calcolo (2009) 46: 157. doi:10.1007/s10092-009-0005-x

Abstract

In this paper we present rigorous a posterioriL2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) TO(1) and Reynolds numbers ν−1≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ν−1≤200.

Keywords

Reduced basis A posteriori error bounds Burgers’ equation Stability factor Successive constraint method Greedy sampling Proper orthogonal decomposition 

Mathematics Subject Classification (2000)

65M12 65M15 

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Ngoc-Cuong Nguyen
    • 1
  • Gianluigi Rozza
    • 1
  • Anthony T. Patera
    • 1
  1. 1.Department of Mechanical EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Ecole Polytechnique Fédérale de LausanneInstitute of Analysis and Scientific ComputingLausanneSwitzerland

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