  This paper introduces a new symbolic-numeric strategy for finding
semidiscretizations of a given PDE that preserve multiple local conservation
laws. We prove that for one spatial dimension, various one-step time
integrators from the literature preserve fully discrete local conservation laws
whose densities are either quadratic or a Hamiltonian. The approach generalizes
to time integrators with more steps and conservation laws of other kinds;
higher-dimensional PDEs can be treated by iterating the new strategy. We use
the Boussinesq equation as a benchmark and introduce new families of schemes of
order two and four that preserve three conservation laws. We show that the new
technique is practicable for PDEs with three dependent variables, introducing
as an example new families of second-order schemes for the potential
Kadomtsev-Petviashvili equation.
