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针对一维浅水方程提出了一种大时间步长格式(LTS),使用这种大时间步长格式可以获得较高的分辨率和计算效率。这种格式最初是由LeVeque提出来的,最近被用于空气动力学的欧拉方程组。在本格式中,传统的近似黎曼求解器被替换成精确黎曼求解器与行波法相结合。当CFL数小于1时,稀疏波采用单波近似就取得了很高的分辨率,但是当CFL数大于1,必须采用两波近似才能保证计算结果是正确的。相比于本格式在空气动力学中的应用,这里采用了更加简化的方法。在空气动力学中,通过等熵流条件来获得稀疏波近似状态变量的值,而在本格式中直接采用波头与波尾的平均。结果证明,采用这种简化对结果没有影响。当CFL数大于10的时候必须对波相撞进行处理以保证结果正确,但是仍然有振荡存在。当CFL数大于15时结果就失真了。通过比较,CFL数在5以下本格式能取得较高的分辨率和计算效率,大于5时,振荡会越来越大,计算效率提高也有限。
A large time-step format (LTS) is proposed for one-dimensional shallow water equations, and a higher resolution and computational efficiency can be obtained using this large time-step format. This format was originally proposed by LeVeque and was recently used in Euler equations for aerodynamics. In this format, the traditional approximate Riemann solver is replaced by the exact Riemann solver combined with the traveling wave method. When the number of CFLs is less than 1, the resolution of the sparse wave obtained by the single-wave approximation is very high. However, when the number of CFLs is greater than 1, the two-wave approximation must be adopted to ensure the calculation result is correct. Compared to the format in the aerodynamic applications, here a more streamlined approach. In aerodynamics, the value of the approximate state variable of the sparse wave is obtained by the isentropic flow conditions, whereas the average of the wave head and the tail is used directly in this format. The results show that using this simplification has no effect on the result. When the CFL is greater than 10, the wave collision must be dealt with to ensure the correct result, but the oscillation still exists. The result is distorted when the number of CFLs is greater than 15. By comparison, the CFL number below 5 in this format can achieve higher resolution and computational efficiency, greater than 5, the oscillation will be more and more, the computational efficiency is also limited.