  Finite difference schemes that preserve two conservation laws of a given
partial differential equation can be found directly by a recently-developed
symbolic approach. Until now, this has been used only for equations with
quadratic nonlinearity. In principle, a simplified version of the direct
approach also works for equations with polynomial nonlinearity of higher
degree. For the modified Korteweg-de Vries equation, whose nonlinear term is
cubic, this approach yields several new families of second-order accurate
schemes that preserve mass and either energy or momentum. Two of these families
contain Average Vector Field schemes of the type developed by Quispel and
coworkers. Numerical tests show that each family includes schemes that are
highly accurate compared to other mass-preserving methods that can be found in
the literature.
