Finite difference can be nonconservative?
Hello
We know that finite volume methods are always locally conservative (my concern is mass only). I was wondering if that is always the case with finite difference methods too? What about for cellcentered finite difference methods in particular. Also, what if the flow being considered is in anisotropic domain, where flux is not necessarily oriented in the direction of gradient. Thanks. 
for incompressible flow, in anycase, it must be conservative. It is not related to the type of discretization.
To solve the incompressible flow, people usually use the fractional step method. (Pressure correction method). At this method, the predictor gives nonconservative velocity field. But it gets corrected by the pressure correction. It is not related to the type of discretization. In all kinds of discretizations, the divergence of the velocity, must be zero. The only thing which must be considered, is the discretization of pressure poisson equation, must adapt the velocity discretization. (this problems arise in unstructured grids). There is another method, namely artificial compressibility method. At this method, pressure correction is not necessary. by adding a term (virtual density) to the continuity equation, and solving the time dependent density. In this case, the velocity is not divergence free per iteration. But to solve time dependent problems, they iterate it per time step, until density converges to the real density. But it is still not related to the type of discretization. Thus I suggest you to control your solver again. 
however, finite difference schemes are not necessarily kinetic energy conserving, especially when going to higher orders. there should be several papers about this topic availables from the Stanford CTR site (by Morinishi and Vasilyev).

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