Model Reduction Techniques for Parametrized Nonlinear Partial Differential Equations

We present model reduction techniques for parametrized nonlinear partial differential equations (PDEs). The main ingredients of our approach are reduced basis (RB) spaces to provide rapidly convergent approximations to the parametric manifold; Galerkin projection of the underlying PDEs onto the RB space to provide reduction in the number of degrees of freedom; and empirical interpolation schemes to provide rapid evaluation of the nonlinear terms associated with the Galerkin projection. We devise a first-order empirical interpolation method to construct an inexpensive and stable interpolation of the nonlinear terms. We consider two different hyper-reduction strategies: hyper-reduction followed by linearization, and linearization followed by hyper-reduction. We extend empirical interpolation to nonintrusive model reduction and apply it to compressible flows in both supersonic and hypersonic regimes. Numerical results are presented to illustrate the accuracy, efficiency, and stability of the reduced-order models.