A systematic method to construct mimetic finite-difference schemes for incompressible flows

We present a general procedure to construct a non-linear mimetic finite-difference operator. The method is very simple and general: it can be applied for any order scheme, for any number of grid points and for any operator constraints. In order to validate the procedure, we apply it to a specific example, the Jacobian operator for the vorticity equation. In particular we consider a finite difference approximation of a second order Jacobian which uses a 9x9 uniform stencil, verifies the skew-symmetric property and satisfies physical constraints such as conservation of energy and enstrophy. This particular choice has been made in order to compare the present scheme with Arakawa’s renowned Jacobian, which turns out to be a specific case of the general solution. Other possible generalizations of Arakawa’s Jacobian are available in literature but only the present approach ensures that the class of solutions found is the widest possible. A simplified analysis of the general scheme is proposed in terms of truncation error and study of the linearised operator together with some numerical experiments. We also propose a class of analytical solutions for the vorticity equation to compare an exact solution with our numerical results.