Spatial Discretization of Partial Differential Equations with Integrals

We consider the problem of constructing spatial finite difference approximations on an arbitrary fixed grid which preserve any number of integrals of the partial differential equation and preserve some of its symmetries. A basis for the space of of such finite difference operators is constructed; most cases of interest involve a single such basis element. (The “Arakawa” Jacobian is such an element, as are discretizations satisfying “summation by parts” identities.) We show how the grid, its symmetries, and the differential operator interact to affect the complexity of the finite difference.