Following the solution formula method given in Dong et al. (High order discontinuities decomposition entropy condition schemes for Euler equations. CFD J. 2002;10(4): 448–457), this article studies a type of one-step fully-discrete scheme, and constructs a third-order scheme which is written into a compact form via a new limiter. The highlights of this study and advantages of new third-order scheme are as follows: ① We proposed a very simple new methodology of constructing one-step, consistent high-order and non-oscillation schemes that do not rely on Runge–Kutta method; ② We systematically studied new scheme's theoretical problems about entropy conditions, error analysis, and non-oscillation conditions; ③ The new scheme achieves exact solution in linear cases and performing better in nonlinear cases when CFL → 1; ④ The new scheme is third order but high resolution with excellent shock-capturing capacity which is comparable to fifth order WENO scheme; ⑤ CPU time of new scheme is only a quarter of WENO5 + RK3 under same computing condition; ⑥ For engineering applications, the new scheme is extended to multi-dimensional Euler equations under curvilinear coordinates. Numerical experiments contain 1D scalar equation, 1D,2D,3D Euler equations. Accuracy tests are carried out using 1D linear scalar equation, 1D Burgers equation and 2D Euler equations and two sonic point tests are carried out to show the effect of entropy condition linearization. All tests are compared with results of WENO5 and finally indicate EC3 is cheaper in computational expense.

中文翻译：

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双曲守恒定律的三阶熵条件格式

遵循 Dong 等人给出的解决方案公式方法。(High order discontinuities Decomposition entropy condition forms for Eulerequations. CFD J . 2002;10(4): 448–457)，本文研究了一类单步全离散格式，并构造了一个三阶格式通过新的限制器写入紧凑的形式。本研究的亮点和新三阶格式的优点如下：①我们提出了一种非常简单的新方法，可以构建不依赖龙格-库塔方法的一步一致的高阶非振荡格式；②系统研究了新方案的熵条件、误差分析、不振荡条件等理论问题；③当CFL→1时，新方案在线性情况下实现了精确解，在非线性情况下表现更好；④ 新方案为三阶，分辨率高，具有优异的冲击捕获能力，可与五阶WENO方案相媲美；⑤ 同等计算条件下，新方案的CPU时间仅为WENO5+RK3的四分之一；⑥ 对于工程应用，新格式被推广到曲线坐标下的多维欧拉方程。数值实验包含一维标量方程、一维、二维、三维欧拉方程。采用一维线性标量方程、一维Burgers方程和二维欧拉方程进行精度测试，并进行两次声波点测试以显示熵条件线性化的效果。所有测试均与 WENO5 的结果进行比较，最终表明 EC3 的计算费用更便宜。