Finite difference schemes with non polynomial local conservation laws

A new technique has been recently introduced to define finite difference schemes that preserve local conservation laws. So far, this approach has been applied to find parametric families of numerical methods with polynomial conservation laws. This paper extends the existing approach to preserve non polynomial conservation laws. Although the approach is general, the treatment of the nonlinear terms depends on the problem at hand. New parameter depending families of conservative schemes are here introduced for the sine-Gordon equation and a magma equation. Optimal methods in each family are identified by finding values of the parameters that minimize a defect-based approximation of the local error in the time discretization.