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

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Abstract

In this paper we present rigorous a posteriori L 2 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) T≈O(1) and Reynolds numbers ν −1≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ν −1≤200.

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Author notes

  1. Gianluigi Rozza

    Present address: Ecole Polytechnique Fédérale de Lausanne, Institute of Analysis and Scientific Computing, Station 8-MA, 1015, Lausanne, Switzerland

Authors and Affiliations

  1. Department of Mechanical Engineering, Massachusetts Institute of Technology, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Ngoc-Cuong Nguyen, Gianluigi Rozza & Anthony T. Patera

Authors
  1. Ngoc-Cuong Nguyen
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  2. Gianluigi Rozza
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  3. Anthony T. Patera
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Correspondence to Gianluigi Rozza.

Additional information

This work was supported by AFOSR Grants FA9550-05-1-0114 and FA-9550-07-1-0425 and the Singapore-MIT Alliance. We acknowledge many helpful discussions with Professor Yvon Maday of University of Paris VI and Dr. Paul Fischer of Argonne National Laboratory and University of Chicago.

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Nguyen, NC., Rozza, G. & Patera, A.T. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo 46, 157–185 (2009). https://doi.org/10.1007/s10092-009-0005-x

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  • DOI: https://doi.org/10.1007/s10092-009-0005-x

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