Abstract: In this paper, an essentially conservative scheme is constructed. The scheme can not be written in the usual conservative form but it can be proven that the limit solutions of this scheme are the weak solutions of hyperbolic conservation laws. It is shown that the scheme is second order accuracy, total variation bounded, stable in L¹ norm and in L^∞ norm, and satisfies the entropy condition.

Key words: essentially conservative scheme, hyperbolic conservation laws, total variation bounded, entropy conditions