Stable and Efficient Galerkin Reduced Order Models for Non-Linear Fluid Flow

An efficient model reduction technique for non-linear compressible flow equations is proposed. The approach is based on the continuous Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. It is an extension of the provablystable model reduction methodology developed previously 1‐3, 13 for the linearized compressible flow equations to the non-linear counterparts of these equations. Attention is focussed on two challenges that arise in developing reduced order models (ROMs) for the full Navier-Stokes equations: stability and efficiency. The former challenge is addressed through the introduction of a transformation into the so-called “entropy variables”. It is shown that performing the Galerkin projection step of the model reduction procedure in these variables leads to a ROM that obeys a priori the second law of thermodynamics, or Clausius-Duhem inequality. In this way, the ROM preserves an essential stability property of the governing equations, that of non-decreasing entropy in the solution. Although the discussion assumes that the reduced basis is constructed via the proper orthogonal decomposition (POD), the entropy stability guarantee holds for any choice of reduced basis, not only the POD basis. The challenge of ensuring that the model reduction technique is efficient in the presence of non-linearities is addressed using the “best points” interpolation method (BPIM) of Peraire, Nguyen et al. 16, 17 To help gauge the viability of the proposed model reduction, some preliminary numerical studies are performed on two non-linear scalar conservation laws whose solutions possess inherently non-linear features, such as shocks and rarefactions: the Burgers equation and the Buckley-Leverett equation.