Mathematical and numerical results on the parametric sensitivity of a ROM-POD of the Burgers equation
Keywords:
ROM, POD, sensitivity, parametric evolution, error estimate, Burgers equationAbstract
We are interested in the mathematical study of the sensitivity of a reduced order model (ROM) of a particular single-parameterised quasi-linear equation, via the parametric evolution. More precisely, the ROM of interest is obtained in two different ways: First, we reduce the complete parametric equation using a proper orthogonal decomposition (POD) basis computed at a given reference value of the parameter, and second the parametric ROM is obtained by an expanded POD basis associated this time to a reference solution and its parametric derivative. The second case of our study was considered in a nearly similar way in Ito and Ravindran (1998), but in the context of the reduced basis (RB) method of the Navier–Stokes equations reduction. Indeed, the authors, Ito and Ravindran (1998) proposed to use an expanded set of basis functions, including solution flows for different values of the Reynolds number and their associated first-order derivatives with respect to this parameter. Beside this work, our second strategy for the parametric ROM-POD construction is to consider a temporal snapshots set including a reference solution and its first-order derivative with respect to the corresponding parameter reference value. We give in both proposed cases of the POD basis construction, an a priori estimate of the parametric squared L2-error between the ROM’s solution and the one associated to the full semi-discrete problem. We will show that this estimate will be depending on the distance between two distinct parameters and the evolution of the ROM’s dimension. Moreover, we show that an a priori upper bound of the squared L2-ROM-POD error is much better in the case of an expanded POD basis functions. In particular, we apply our theoretical study to the one-dimensional Burgers equation. Numerical tests are done for the one-dimensional Burgers equation, only in the case of a POD basis associated with a reference solution at a fixed value of the viscosity.
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