  DiPerna-Lions (Invent. Math. 1989) established the existence and uniqueness
results for linear transport equations with Sobolev velocity fields. This paper
provides mathematical analysis on two simple finite difference methods applied
to linear transport equations on a bounded domain with divergence-free
(unbounded) Sobolev velocity fields. The first method is based on a
Lax-Friedrichs type explicit scheme with a generalized hyperbolic scale, where
truncation of an unbounded velocity field and its measure estimate are
implemented to ensure the monotonicity of the scheme; the method is
$L^p$-strongly convergent. The second method is based on an implicit scheme
with $L^2$-estimates, where the discrete Helmholtz-Hodge decomposition for
discretized velocity fields plays an important role to ensure the
divergence-free constraint in the discrete problem; the method is scale-free
and $L^2$-strongly convergent. The key point for both of our methods is to
obtain fine $L^2$-bounds of approximate solutions that tend to the norm of the
exact solution given by DiPerna-Lions. Finally, the explicit scheme is applied
to the case with smooth velocity fields from the viewpoint of the level-set
method involving transport equations, where rigorous discrete approximation of
geometric quantities of level sets is discussed.
