  A novel numerical strategy is introduced for computing approximations of
solutions to a Cahn-Hilliard model with degenerate mobilities. This model has
recently been introduced as a second-order phase-field approximation for
surface diffusion flows. Its numerical discretization is challenging due to the
degeneracy of the mobilities, which generally requires an implicit treatment to
avoid stability issues at the price of increased complexity costs. To mitigate
this drawback, we consider new first- and second-order Scalar Auxiliary
Variable (SAV) schemes that, differently from existing approaches, focus on the
relaxation of the mobility, rather than the Cahn-Hilliard energy. These schemes
are introduced and analysed theoretically in the general context of gradient
flows and then specialised for the Cahn-Hilliard equation with mobilities.
Various numerical experiments are conducted to highlight the advantages of
these new schemes in terms of accuracy, effectiveness and computational cost.
