  This work proposes a method for model reduction of finite-volume models that
guarantees the resulting reduced-order model is conservative, thereby
preserving the structure intrinsic to finite-volume discretizations. The
proposed reduced-order models associate with optimization problems
characterized by a minimum-residual objective function and nonlinear equality
constraints that explicitly enforce conservation over subdomains. Conservative
Galerkin projection arises from formulating this optimization problem at the
time-continuous level, while conservative least-squares Petrov--Galerkin (LSPG)
projection associates with a time-discrete formulation. We equip these
approaches with hyper-reduction techniques in the case of nonlinear flux and
source terms, and also provide approaches for handling infeasibility. In
addition, we perform analyses that include deriving conditions under which
conservative Galerkin and conservative LSPG are equivalent, as well as deriving
a posteriori error bounds. Numerical experiments performed on a parameterized
quasi-1D Euler equation demonstrate the ability of the proposed method to
ensure not only global conservation, but also significantly lower state-space
errors than nonconservative reduced-order models such as standard Galerkin and
LSPG projection.
