  Finite-dimensional truncations are routinely used to approximate partial
differential equations (PDEs), either to obtain numerical solutions or to
derive reduced-order models. The resulting discretized equations are known to
violate certain physical properties of the system. In particular, first
integrals of the PDE may not remain invariant after discretization. Here, we
use the method of reduced-order nonlinear solutions (RONS) to ensure that the
conserved quantities of the PDE survive its finite-dimensional truncation. In
particular, we develop two methods: Galerkin RONS and finite volume RONS.
Galerkin RONS ensures the conservation of first integrals in Galerkin-type
truncations, whether used for direct numerical simulations or reduced-order
modeling. Similarly, finite volume RONS conserves any number of first integrals
of the system, including its total energy, after finite volume discretization.
Both methods are applicable to general time-dependent PDEs and can be easily
incorporated in existing Galerkin-type or finite volume code. We demonstrate
the efficacy of our methods on two examples: direct numerical simulations of
the shallow water equation and a reduced-order model of the nonlinear
Schrodinger equation. As a byproduct, we also generalize RONS to phenomena
described by a system of PDEs.
