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 Ismails 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 Roes algorithm, the central upwind schemes and Ismails 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.

Key words: hyperbolic conservation law equation, semi-discrete method, fourth-order scheme, optimal Runge-Kutta method

摘要： 针对一维Burgers方程和一维Euler方程组的数值求解问题,提出了一种四阶高分辨率熵相容算法。新算法时间方向采用半离散方式,空间方向应用四阶中心加权基本无振荡(CWENO)重构方法,数值通量引入Ismail通量函数,将新的四阶算法应用于静态激波问题、激波管问题以及强稀疏波问题的数值求解中,并将所得结果同准确解以及已有算法所得结果进行了分析与比较。数值结果表明:新算法计算结果正确、分辨率高,能够准确捕捉激波及稀疏波，并能有效避免膨胀激波的产生。新算法适用于准确解决一维Burgers方程和一维Euler方程组的数值求解问题。

关键词: 双曲守恒律方程, 半离散方法, 四阶格式, 优化龙格-库塔法