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

Abstract

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%.