Windowed space-time least-squares Petrov-Galerkin model order reduction for nonlinear dynamical systems

Projection-based reduced-order models (PROMs) restrict the full-order model (FOM) to a low-dimensional subspace. Space-time PROMs in particular restrict the FOM to a temporally space-time trial subspace, and compute approximate solutions through a residual orthogonalization or minimization process. One such technique of interest is the space- time least-squares Petrov-Galerkin method (ST-LSPG), which reduces both the spatial and temporal dimensions. However, ST-LSPG is computationally expensive, because it requires solving a dense space-time system with a space-time basis that is calculated over the entire global time domain, which can be unfeasible for large-scale applications. To address these challenges, this paper presents the windowed space-time least-squares Petrov-Galerkin method (WST-LSPG) for model reduction of nonlinear parameterized dynamical systems. The proposed WST-LSPG approach addresses the aforementioned challenges by (1) dividing the time simulation into windows, (2) devising a unique low-dimensional high-fidelity space-time trial subspace for each window, and (3) minimizing the discrete-in-time space-time residual of the dynamical system over each window. In this formulation, the problem confines coupling within each window, but solves space-time residual minimization problems sequentially across the windows. WST-LSPG is equipped with hyper-reduction techniques to further reduce the computational cost. Numerical experiments for the one-dimensional Burgers' equation and the two-dimensional compressible Navier-Stokes equations for flow over a NACA 0012 airfoil demonstrate that WST-LSPG is superior to ST-LSPG in terms of accuracy and computational gain by as much as 99%. Published by Elsevier B.V.