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A proper orthogonal decomposition (POD) method is used to interpolate the flow around an airfoil for various Mach numbers and angles of attack in the transonic regime. POD uses a few numerical simulations, called snapshots, to create eigenfunctions. These eigenfunctions are combined using weighting coefficients to create a new solution for different values of the input parameters. Since POD methods are linear, their interpolation capabilities are quite limited when dealing with flow presenting nonlinearities, such as shocks. In order to improve their performance for cases involving shocks, a new method is proposed using variable fidelity. The main idea is to use POD to interpolate the difference between the CFD solution obtained on two different grids, a coarse one and a fine one. Then, for any new input parameter value, a coarse grid solution is computed using CFD and the POD interpolated difference is added to predict the fine grid solution. This allows some nonlinearities associated with the flow to be introduced. Results for various Mach numbers and angles of attack are compared to full CFD results. The variable fidelity-based POD method shows good improvement over the classical approach.
Malouin et al. (2013) studied this question.