Dissipative/conservative Galerkin method using discrete partial derivatives for nonlinear evolution equations

A new method is proposed for designing Galerkin schemes that retain the energy dissipation or conservation properties of nonlinear evolution equations such as the Cahn–Hilliard equation, the Korteweg–de Vries equation, or the nonlinear Schrödinger equation. In particular, as a special case, dissipative or conservative finite-element schemes can be derived. The key device there is the new concept of discrete partial derivatives. As examples of the application of the present method, dissipative or conservative Galerkin schemes are presented for the three equations with some numerical experiments.