In recent years, Mao and his co-workers developed a class of finite-volume schemes for evolution partial differential equations, see [1-5].The schemes show a super-convergence quality and have good structure-preserving property in long-time numerical simulations. In , Chen and Ma developed a scheme which combine the idea of paper  and that of the the second-order ENO scheme . In this paper, we propose a scheme which extend the result of  and obtain the scheme using the third-order ENO reconstruction. Numerical experiments show that our scheme is robust in long-time behaviors. Numerical solutions are far better than those of .