TY - JOUR
T1 - A Systematic Method to Construct Mimetic Finite-Difference Schemes for Incompressible Flows.
AU - Chiara Sorgentone & Bernardo Favini
JO - International Journal of Numerical Analysis and Modeling
VL - 3
SP - 419
EP - 436
PY - 2017
DA - 2017/06
SN - 14
DO - http://dor.org/
UR - https://global-sci.org/intro/article_detail/ijnam/10015.html
KW - Mimetic schemes
KW - Arakawa's Jacobian
KW - finite-difference
KW - non-linear instability
AB - 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 9× 9 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.