任意线性迎风格式在固定网格中的自由流保持
On free-stream preservation in stationary grids for arbitrary linear upwind schemes
摘要：
为了提高高阶差分格式的应用成熟度，自由流保持性近年来已被广泛认识，成为研究的热点。在过去的文献中，只有中心格式被认为是适合于自由流保持。在这项研究中，任意线性计划，以实现财产的方法进行了研究。首先回顾了Thomas、Lombard和Neier对网格度规的推导，并通过Vinokur和Yee的证明，首次证明了度规和不可分裂通量的线性格式可以得到该性质。在实际应用中，通量分裂的存在使得直接使用迎风格式难以实现自由流的保持。为了克服这一困难，进行了两项尝试：首先，进行中心格式分解，通过中心格式分解推导出中心差分格式来逼近度量计算中的一阶偏导数;其次，提出了通量分裂的处理方法，并给出了具体例子。通过这两种尝试，线性迎风节点格式可以实现该性质。对于半节点或混合型格式，应使用插值来导出半节点处的变量。因此，方向一致的插值建议，这是必要的，以避免违反的度量身份和自由流的保护。还测试了两个数值问题，即，波浪形、大部分随机化和三角形网格上的自由流和涡流保持。数值结果验证了上述理论结果，特别是三角形网格上的模拟结果表明，本文所讨论的方法是结构网格上的典型算法，对非结构网格上的问题具有应用潜力。
英文摘要：
In order to improve the application maturity of high-order difference schemes, the free-stream preservation property, whose importance has been widely recognized in recent years, has been developed into a focus of study.. In past literatures, only central schemes are considered to be suitable for free-stream preservation. In this study, the methodology for arbitrary linear schemes to achieve the property is investigated. First, derivations of grid metric by Thomas, Lombard and Neier are reviewed, through which linear schemes for the metric and unsplit flux could attain the property by the proof of Vinokur and Yee firstly. In practical applications, flux splittings are usually at presence and therefore the direct use of upwind schemes seems difficult to fulfill free-stream preservation. To overcome the difficulty, two attempts are made: firstly, a central-scheme-decomposition is worked out, through which a central difference scheme is derived to approximate the first-order partial derivative in metric evaluations; secondly, treatments are proposed for flux splitting and a concrete example is presented. Through these two attempts, the linear upwind node schemes can achieve the property. For half-node or mixed type schemes, interpolations should be used to derive variables at half nodes. As a result, a directionally consistent interpolation is proposed, which is shown to be necessary in order to avoid violation of the metric identity and free-stream preservation. Two numerical problems are also tested, i.e., the free-stream and vortex preservation on wavy, largely randomized and triangular grids. Numerical results validate aforementioned theoretical outcomes; especially, simulations on the triangular grids indicate the methods discussed in this study, which are typical algorithms on structured grids, has the application potential for problems on unstructured meshes.
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