  A classical reduced order model (ROM) for dynamical problems typically
involves only the spatial reduction of a given problem. Recently, a novel
space-time ROM for linear dynamical problems has been developed, which further
reduces the problem size by introducing a temporal reduction in addition to a
spatial reduction without much loss in accuracy. The authors show an order of a
thousand speed-up with a relative error of less than 0.00001 for a large-scale
Boltzmann transport problem. In this work, we present for the first time the
derivation of the space-time Petrov-Galerkin projection for linear dynamical
systems and its corresponding block structures. Utilizing these block
structures, we demonstrate the ease of construction of the space-time ROM
method with two model problems: 2D diffusion and 2D convection diffusion, with
and without a linear source term. For each problem, we demonstrate the entire
process of generating the full order model (FOM) data, constructing the
space-time ROM, and predicting the reduced-order solutions, all in less than
120 lines of Python code. We compare our Petrov-Galerkin method with the
traditional Galerkin method and show that the space-time ROMs can achieve
O(100) speed-ups with O(0.001) to O(0.0001) relative errors for these problems.
Finally, we present an error analysis for the space-time Petrov-Galerkin
projection and derive an error bound, which shows an improvement compared to
traditional spatial Galerkin ROM methods.
