  We present a parametric family of semi-implicit second order accurate
numerical methods for non-conservative and conservative advection equation for
which the numerical solutions can be obtained in a fixed number of forward and
backward alternating substitutions. The methods use a novel combination of
implicit and explicit time discretizations for one-dimensional case and the
Strang splitting method in several dimensional case. The methods are described
for advection equations with a continuous variable velocity that can change its
sign inside of computational domain. The methods are unconditionally stable in
the non-conservative case for variable velocity and for variable numerical
parameter. Several numerical experiments confirm the advantages of presented
methods including an involvement of differential programming to find optimized
values of the variable numerical parameter.
