Abstract：The fourth-order entropy consistent schemes were proposed for one-dimensional Burgers equation and one-dimensional Euler systems. Semi-discrete method was used in time, the fourth-order Central Weighted Essentially Non-Oscillatory (CWENO) reconstruction was utilized in space and the Ismails numerical flux function was introduced for the new algorithm. The new scheme was applied for the static shock wave, the Sod shock tube and strong rarefaction wave problems. The numerical results were compared with their corresponding exact solutions and the other existing algorithms results. According to the results, this new method has higher resolution than Roes algorithm, the central upwind schemes and Ismails method have. Moreover, the new algorithm can accurately capture the shock waves and the rarefaction waves without non-physical oscillations. In a word, it is a feasible and accurate numerical method for one-dimensional Burgers equation and one-dimensional Euler systems.