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首页> 外文期刊>Journal of the Brazilian Society of Mechanical Sciences and Engineering >Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes
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Numerical solution of the Falkner-Skan equation using third-order and high-order-compact finite difference schemes

机译：使用三阶和高阶紧致有限差分格式的Falkner-Skan方程数值解

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摘要

We present a computational study of the solution of the Falkner-Skan equation (a thirdorder boundary value problem arising in boundary-layer theory) using high-order and high-order-compact finite differences schemes. There are a number of previously reported solution approaches that adopt a reduced-order system of equations, and numerical methods such as: shooting, Taylor series, Runge-Kutta and other semi-analytic methods. Interestingly, though, methods that solve the original non-reduced third-order equation directly are absent from the literature. Two high-order schemes are presented using both explicit (third-order) and implicit compact-difference (fourth-order) formulations on a semi-infinite domain; to our knowledge this is the first time that high-order finite difference schemes are presented to find numerical solutions to the non-reduced-order Falkner-Skan equation directly. This approach maintains the simplicity of Taylor-series coefficient matching methods, avoiding complicated numerical algorithms, and in turn presents valuable information about the numerical behaviour of the equation. The accuracy and effectiveness of this approach is established by comparison with published data for accelerating, constant and decelerating flows; excellent agreement is observed. In general, the numerical behaviour of formulations that seek an optimum physical domain size (for a given computational grid) is discussed. Based on new insight into such methods, an alternative optimisation procedure is proposed that should increase the range of initial seed points for which convergence can be achieved.
机译：我们使用高阶和高阶紧致有限差分方案对Falkner-Skan方程（边界层理论中出现的三阶边值问题）的解决方案进行了计算研究。有许多以前报告的解决方案方法采用了降阶方程组和数值方法，例如：射击，泰勒级数，朗格-库塔（Runge-Kutta）和其他半解析方法。有趣的是，文献中缺少直接求解原始的非约化三阶方程的方法。在半无限域上使用显式（三阶）和隐式紧致差分（四阶）公式，提出了两种高阶方案。据我们所知，这是首次提出高阶有限差分方案以直接找到非降阶Falkner-Skan方程的数值解。这种方法保持了泰勒级数系数匹配方法的简单性，避免了复杂的数值算法，从而提供了有关方程数值行为的有价值的信息。这种方法的准确性和有效性是通过与已发布的用于加速，恒定和减速流量的数据进行比较来确定的;观察到极好的一致性。通常，讨论寻求最佳物理域大小（对于给定的计算网格）的制剂的数值行为。基于对此类方法的新见识，提出了一种替代性优化程序，该程序应增加可以实现收敛的初始种子点的范围。

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