**1. Introduction**

During the past few decades, abundant numerical methods for solving hyperbolic conservation laws have been designed; one can consult the review papers [1,2] and the references therein. Among the various methods, high order schemes, such as total variation diminishing (TVD) schemes, weighted essentially non-oscillatory (WENO) schemes and discontinuous Galerkin (DG) schemes, have achieved great success. However, it seems that there are few entropy stability results for high order numerical schemes for hyperbolic conservation laws, especially for the nonlinear hyperbolic conservation laws. In view of the above discussion, we limit our research on high order entropy stable methods for solving hyperbolic conservation laws.

In accordance with the idea of Tadmor, entropy schemes are often constructed by utilizing entropy conservative flux in conjunction with suitable numerical diffusion [3]. For example, physical viscosity was discretized and used as numerical diffusion in [4]. Roe-type numerical diffusion was selected for various systems like Euler equations, shallow water equations and ideal magnetohydrodynamic equations [5–7]. Based on the limiter mechanism, Liu et al. constructed a family of entropy consistent schemes with high resolution [8]. Recently, Dubey discussed the amount of suitable diffusion for the sake of devising non-oscillatory entropy stable schemes in the TVD sense [9]. To obtain high order entropy stable schemes, Fjordholm et al. firstly proved that ENO reconstruction satisfies a sign property and presented entropy stable schemes based on ENO reconstruction of arbitrary order accuracy [10,11]. To overcome the drawbacks of ENO reconstruction, we presented a third order reconstruction which is non-oscillatory and satisfies the sign property to construct a third order entropy stable scheme [12]. WENO reconstruction was modified to satisfy the sign property and also used to construct an entropy stable scheme in [13,14], although they are limited to third order accuracy.

In this paper, we are aiming at presenting a new entropy stable scheme of fourth order accuracy to solve hyperbolic conservation laws in one dimension. First, we employ a fourth order entropy conservative flux which is a linear combination of two-point entropy conservative fluxes. Then, we present a non-oscillatory reconstruction of fourth order accuracy which possesses the sign property to obtain a fourth order accurate numerical diffusion operator. By adding this numerical diffusion operator to the entropy conservative flux, the resulting flux is entropy stable.

The remainder of this paper is given as follows. Section 2 describes the procedure for building our fourth order entropy scheme. After that, we present some typical numerical experiments to show the effectiveness of the newly developed scheme. Finally, concluding remarks are given in Section 4.
