  Central-upwind (CU) schemes are Riemann-problem-solver-free finite-volume
methods widely applied to a variety of hyperbolic systems of PDEs. Exact
solutions of these systems typically satisfy certain bounds, and it is highly
desirable or even crucial for the numerical schemes to preserve these bounds.
In this paper, we develop and analyze bound-preserving (BP) CU schemes for
general hyperbolic systems of conservation laws. Unlike many other Godunov-type
methods, CU schemes cannot, in general, be recast as convex combinations of
first-order BP schemes. Consequently, standard BP analysis techniques are
invalidated. We address these challenges by establishing a novel framework for
analyzing the BP property of CU schemes. To this end, we discover that the CU
schemes can be decomposed as a convex combination of several intermediate
solution states. Thanks to this key finding, the goal of designing BPCU schemes
is simplified to the enforcement of four more accessible BP conditions, each of
which can be achieved with the help of a minor modification of the CU schemes.
We employ the proposed approach to construct provably BPCU schemes for the
Euler equations of gas dynamics. The robustness and effectiveness of the BPCU
schemes are validated by several demanding numerical examples, including
high-speed jet problems, flow past a forward-facing step, and a shock
diffraction problem.
