  This paper establishes a structure-preserving numerical scheme for the
Cahn--Hilliard equation with degenerate mobility. First, by applying a finite
volume method with upwind numerical fluxes to the degenerate Cahn--Hilliard
equation rewritten by the scalar auxiliary variable (SAV) approach, we
creatively obtain an unconditionally bound-preserving, energy-stable and
fully-discrete scheme, which, for the first time, addresses the boundedness of
the classical SAV approach under $H^{-1}$-gradient flow. Then, a
dimensional-splitting technique is introduced in high-dimensional cases, which
greatly reduces the computational complexity while preserves original
structural properties. Numerical experiments are presented to verify the
bound-preserving and energy-stable properties of the proposed scheme. Finally,
by applying the proposed structure-preserving scheme, we numerically
demonstrate that surface diffusion can be approximated by the Cahn--Hilliard
equation with degenerate mobility and Flory--Huggins potential when the
absolute temperature is sufficiently low, which agrees well with the
theoretical result by using formal asymptotic analysis.wn theoretically by
formal matched asymptotics.
