  Godunov type numerical schemes for the class of hyperbolic systems, admitting
non-classical $\delta-$ shocks are proposed. It is shown that the numerical
approximations converge to the solution and preserve the physical properties of
the system such as positive density and bounded velocity. The scheme has been
extended to positivity preserving and velocity bound preserving second-order
accurate scheme by using appropriate slope limiters. The numerical results are
compared with the existing the literature and the scheme is shown to capture
the solution efficiently. The paper presents a hyperbolic system, for which an
entropy satisfying scheme is constructed through an appropriate decoupling of
the system into two scalar conservation laws with discontinuous flux.
