Energy method for structure-preserving finite difference schemes and some properties of difference quotient

In this article we introduce the energy method for structure-preserving finite difference schemes which inherit the physical structures such as energy conservation or dissipative laws. Another aim is to give some useful properties for difference quotient which is compatible with the structure-preserving finite difference schemes. The method and properties enable us not only to take the problem with more general nonlinearity but also to improve proofs of error estimate between the numerical and exact solutions. Lastly we give two examples of application to the Cahn–Hilliard and the Boussinesq type equations.