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Spatially Accurate Conservative or Dissipative Finite Difference Schemes Derived by the Discrete Variational Method

A method, called “the discrete variational method”, has been recently presented by Furihata and Matsuo for designing finite difference schemes that inherit energy conservation or dissipation property from nonlinear partial differential equations (PDEs). In this paper the method is enhanced so that the derived schemes be highly accurate in space by introducing higher order spatial difference operators, including the so-called “spectral differentiation” operator. Applications to the KdV equation, and the cubic nonlinear Schrödinger equation are also presented.