Space-time model reduction using Legendre-POD modes

Space-time model reduction involves solving a set of algebraic equations for static coefficients that collectively represent a trajectory over the interval [0,T]. An advantage compared to space-only model reduction, where coefficients representing the instantaneous state are integrated forward in time, is that the space-time representation of a trajectory is substantially more efficient because it leverages spatiotemporal correlations. For (asymptotically) short time intervals, the optimal space-time basis vectors are Legendre proper orthogonal decomposition (LPOD) modes — modes with Legendre temporal dependence and spatial dependence tailored to each polynomial. In this work, we derive a space-time reduced-order model (ROM) that computes the LPOD coefficients for the interval [0,T] given the initial condition and system forcing. This is accomplished by first considering a linearized system and then projecting the linearized-system solution operators onto the LPOD modes. We explore obtaining the projections of these operators by both analytic and data-driven means. Nonlinearity is handled by computing the nonlinearity that would result from the solution to the linearized system, treating this nonlinearity as an additional forcing on the system, and iterating this process until convergence. We find that the ROM yields errors comparable to the projection of the full-order solution onto the LPOD modes, which is a lower bound for the approach. We also compare against standard ROM benchmarks.