【作者】 杨水平；

【导师】 李寿佛；

【作者基本信息】 湘潭大学 ， 计算数学， 2006， 硕士

【摘要】 由于求解双曲守恒律组的高阶加权实质上无振荡有限差分格式(简记为FD-WENO)是最近十年才发展起来的，它与上世纪70-80年代发展起来的著名高阶Godunov格式比较，究竟优缺点各如何，哪种格式更好，这个问题目前在国内外文献中研究尚少，尚无定评。特别是对于辐射流体力学问题的数值计算，以上两种格式中究竟哪种格式的实际效果更好，这是惯性约束聚变(ICF)数值模拟研究中迫切需要回答的问题，为此，本文作者发展了用2阶Godunov格式MUSCL求解气体动力学Euler方程的应用软件，并通过同时用五阶FD-WENO格式(下文中简记为WENO5)和二阶MUSCL格式求解各种有代表性的流体力学问题，进行了大量的数值测试，例如，通过求解线性初边值问题、一维Burgers方程初边值问题、若干Riemann问题、一维激波相互碰撞问题及二维周期漩涡问题等对这两种格式进行了测试和定量比较，通过求解Richtmyer-Meshkov(RM)不稳定性问题和Rayleigh-Taylor(RT)不稳定性问题对这两种格式进行了测试和定性比较．我们发现对于简单的Sod Riemann问题或Lax Riemann问题，两种格式都易于算出令人满意的具有较高精度和较高分辨率的数值结果，相对来说，WENO5得到的数值结果粘性稍强，MUSCL得到的结果则在局部地方有微小的非物理振荡，但计算速度更快；对于Peak Riemann问题及Woodward-Colella激波相互碰撞问题，WENO5所获数值结果在激波及稀疏波附近明显优于MUSCL，前者不仅分辨率较高，误差也较小；对于二维周期漩涡问题，采用81×81的一致网格用WENO5格式计算至时刻t=10的数值结果甚至比采用161×161的一致网格用MUSCL格式计算至时刻t=10的数值结果要好得多，但前者花费的计算时间更短；对于RM不稳定性问题和RT不稳定性问题，WENO5所得到的数值结果比MUSCL所得到的数值结果分辨率更高，图形较为符合实际情况。本文的研究结果表明，尽管对于简单的Riemann问题，二阶MUSCL具有一定优势，但对于既具有强激波又具有大面积复杂流动结构的问题，特别是对于ICF数值模拟问题，二阶格式一般难以提供具有可接受分辨率的数值结果，而高阶FD-WENO格式特别适合于求解这类问题。更多还原

【Abstract】 A numerical study is undertaken comparing the fifth-order finite difference weighted essentially non-oscillatory scheme (FD-WEN05) to the second-order Godunov scheme MUSCL (monotonic upstream-centered scheme for conservation laws). For quantitative comparison purpose, these methods are tested on a series of problems whose true solutions are known or can be computed with high accuracy, such as linear advection problem, Sod Riemann, "peak" Riemann problem, Woodward and Colella interacting shock wave problem and the two-dimensional periodic vortex problem. For qualitative comparison purpose, these methods are tested on Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems. The numerical results show that: for the Sod and Lax Riemann problems, high accuracy and high resolution results can be easily achieved by both schemes, but MUSCL is performed much faster than WEN05. For the " Peak" Riemann problem and the Woodward and Colella interacting shock wave problem, when both methods are performed on same space mesh, we find that WENO5 is better than MUSCL around the shock and the rarefaction, in more de-tail, WENO5 has not only higher resolution but also smaller errors than MUSCL. For the two-dimensional periodic vortex problem, We find that the numerical results of WENO5 when performed on the uniform mesh of 81 x 81 points are much more accurate than that of MUSCL when performed on the refined uniform mesh of 161 ×161 points, and the former takes CPU time shorter than the latter. For Rayleigh-Taylor instability and Richtmyer-Meshkov instability problems, when both methods are performed on same space mesh, we find that the numerical results of WENO5 are rich of higher resolution and the images are more realistic. In view of all the results mentioned above, we thus conclude that the second-order Godunov scheme MUSCL has advantages for many problems containing only simple shocks with almost linear smooth solutions in between, such as the solutions to most Riemann problems. However, when the solution contains both discontinuities and complex solution structures in the smooth regions, a higher order method, such as WENO5, may be more realistic and more economical in CPU time.更多还原