Nonlinear model reduction for unsteady discontinuous flows

We develop a new nonlinear reduced-order model (ROM) based on proper orthogonal decomposition (POD), which can be used for quantitative simulation of not only smooth flows, but also flows with strong discontinuities. The new model is derived using a Galerkin projection of the fully conservative, nonlinear discretized 2-D Euler equations onto the POD basis constructed for each conservative variable. This approach can be interpreted as a variant of the spectral method with a truncated set of basis functions. A system of ordinary differential equations (ODEs) derived using this model reduction technique resembles the major nonlinear and conservation properties of the original discretized Euler equations. The new reduced-order model also preserves the stability properties of the discrete full-order model equations, so that no additional stabilization is required unlike conventional POD-based models that are susceptible to numerical instabilities. The performance of the new POD ROM is evaluated for 2-D compressible unsteady inviscid flows over a wide range of Mach numbers including trans- and supersonic flows with strong shock waves.