  Least-squares Petrov--Galerkin (LSPG) model-reduction techniques such as the
Gauss--Newton with Approximated Tensors (GNAT) method have shown promise, as
they have generated stable, accurate solutions for large-scale turbulent,
compressible flow problems where standard Galerkin techniques have failed.
However, there has been limited comparative analysis of the two approaches.
This is due in part to difficulties arising from the fact that Galerkin
techniques perform optimal projection associated with residual minimization at
the time-continuous level, while LSPG techniques do so at the time-discrete
level. This work provides a detailed theoretical and computational comparison
of the two techniques for two common classes of time integrators: linear
multistep schemes and Runge--Kutta schemes. We present a number of new
findings, including conditions under which the LSPG ROM has a time-continuous
representation, conditions under which the two techniques are equivalent, and
time-discrete error bounds for the two approaches. Perhaps most surprisingly,
we demonstrate both theoretically and computationally that decreasing the time
step does not necessarily decrease the error for the LSPG ROM; instead, the
time step should be `matched' to the spectral content of the reduced basis. In
numerical experiments carried out on a turbulent compressible-flow problem with
over one million unknowns, we show that increasing the time step to an
intermediate value decreases both the error and the simulation time of the LSPG
reduced-order model by an order of magnitude.
