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

By Ngoc Cuong Nguyen, Gianluigi Rozza, Anthony T. Patera. 2009

149 citations recorded by the source corpus.

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) 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.

Subfields: Computational Mechanics, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/298c9c6bbf55de4b9bdee3e7d9a2043d32d526b1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Numerical Analysis

In Statistical and Nonlinear Physics

Citations in the corpus

An hp certified reduced basis method for parametrized parabolic partial differential equations (2011, unknown): “This class of problems is particularly ‘ripe’ for the hp approach due to the O(N4) computational cost associated with RB error bound evaluation [11,12]: even a small reduction in N – the number of RB basis functions – will result in significant Online computational savings.”
Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Parabolic PDEs: Application to Real‐Time Bayesian Parameter Estimation (2010, background): “In particular, it can readily be proven (Grepl and Patera 2005; Haasdonk and Ohlberger 2008; Nguyen et al .”
Certified reduced basis approximation for parametrized partial differential equations and applications (2011, methodology): “An efficient combination of the two techniques greedy-POD in parameter-time has been proposed [22,23] and is currently used for the treatment of parabolic problems [24]; see Sect.”
Stabilized reduced basis method for parametrized advection-diffusion PDEs (2014, methodology): “More precisely, we construct the reduced basis in the time-dependent case following the so called POD-Greedy approach [11, 21, 22, 25].”
REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS (2011, methodology): “Our main intention here is to recall the operation counts for the Construction and Evaluation stages; we refer the reader to the discussion in [44] for further details.”
Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations (2013, methodology): “It is worth mentioning here that the reduced basis method stu ied in [13] has been applied to the unsteady Burgers’ equation in one space dimension and a posteriorierror bounds were derived for the approximation.”
Model Order Reduction in Fluid Dynamics: Challenges and Perspectives (2014, methodology): “We start by recalling that current approaches for constructing reduced basis approximations of time-dependent parametrized PDEs exploit a combined POD-greedy procedure – POD in time to capture the causality associated with the evolution equation, greedy procedure for sampling the parameter space and”
CERTIFIED REDUCED BASIS METHODS FOR NONAFFINE LINEAR TIME-VARYING AND NONLINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS (2012, background): “In the recent past, reduced basis approximations and associated a posteriori error estimation for linear and at most quadratically nonlinear elliptic and parabolic PDEs honoring this requirement have been successfully developed.(19,20,23,32,30,36,46,48,49) In Ref.”
Nonlinear Model Reduction via Discrete Empirical Interpolation (2010, background): “Its success is limited to the problems of linear elliptic parabolic PDEs with affine parameters or low-order polynomial nonlinearities [32, 25, 26, 41, 29].”
Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem (2011, background): “1, but the basis functions themselves are obtained by performing a POD (in time) for this identified parameter (see [15,16]).”