Shape preserving schemes

The problem with this particular duality is that the dual of an inflexion in a curve is an asymptote, so that if the polygon has an inflexion the curve will contain pieces of hyperbola going off to infinity. If rules are put in place that something special is done when the polygon has an inflexion, however, this can give a nice shape-preserving scheme which has no inflexions in the limit curve when the initial polygon has none. 

Numerical methods constructed based on the advective form (non-conservative form) of the transport operator are shape preserving, but not conservative [82]. Schemes constructed based on the conservative form (or flux form) of the transport operator are preferable when strict conservation is required. 

Thuburn J (1995) Dissipation and Cascades to Small Scales in Numerical Models Using a Shape-Preserving Advection Scheme. Mon Wea Rev 123(6) 1888-1903... 

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