  Symmetry- and conservation law-preserving finite difference discretizations
are obtained for linear and nonlinear one-dimensional wave equations on five-
and nine-point stencils, using the theory of Lie point symmetries of difference
equations, and the discrete direct multiplier method of conservation law
construction. In particular, for the linear wave equation, an explicit
five-point scheme is presented that preserves the discrete analogs of its basic
geometric point symmetries, and six of the corresponding conservation laws. For
a class of nonlinear wave equations arising in hyperelasticity, a nine-point
implicit scheme is constructed, preserving four point symmetries and three
local conservation laws. Other discretization of the nonlinear wave equations
preserving different subsets of conservation laws are discussed.
