, Volume 46, Issue 3, pp 157-185
Date: 30 Jun 2009

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) 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.

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.