In this paper, we investigate a well-balanced and positive-preserving
non-staggered central scheme, which has second-order accuracy on both
time and spatial scales, for open channel flows with variable channel
width and non-flat bottom. We perform piecewise linear reconstructions
of the conserved variables and energy as well as discretize the source
term using the property that the energy remains constant, so that the
complex source term and the flux can be precisely balanced so as to
maintain the steady state. The scheme also ensures that the
cross-sectional wet area is positive by introducing a draining time-step
technique. Numerical experiments demonstrate that the scheme is capable
of accurately maintaining both the still steady-state solutions and the
moving steady-state solutions, simultaneously. Moreover, the scheme has
the ability to accurately capture small perturbations in the moving
steady-state solution and avoid generating spurious oscillations. It is
also capable of showing that the scheme is positive-preserving and
robust in solving the dam-break problem.