A Fourth-Order Symmetric WENO Scheme with Improved Performance by New Linear and Nonlinear Optimizations
A Fourth-Order Symmetric WENO Scheme with Improved Performance by New Linear and Nonlinear Optimizations
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DOI：
10.1007/s10915-016-0293-7
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发表时间：
2017-04
影响因子：
2.5
通讯作者：
Qin Li;Qilong Guo;Dong Sun;Pengxin Liu;Hanxin Zhang
中科院分区：
文献类型：
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作者：
Qin Li;Qilong Guo;Dong Sun;Pengxin Liu;Hanxin Zhang
提出了一个四阶、对称、加权的本质无振荡格式，该格式具有改进的线性和非线性性质。为了改善其色散和耗散关系的线性性质，本文提出了一种通过求解方程组来优化的方法。使用这种方法，精心选择的优化目标实现，通过它获得了改善的线性性能，同时保持稳定的冲击反射问题。非线性格式的实现是另一个重要的问题，它通常会影响其在实际应用中的性能。在这方面的优化被认为是使该计划的工作在其线性形式有效和问题独立的流量变化相对平稳时。为了实现这一目标，研究了四个主题。第一种是基于全变分概念的新的平滑度混合指标，通过该指标可以获得较好的分辨率，用于分辨短波长的结构。第二种是专门为最顺风指示器设计的修改，以避免数值振荡。第三种是一种新的转换算法，通过一个变量使该方案在其线性和非线性状态之间工作。新算法的设计避免了对光滑流场和振荡流场的误判。第四种是关于案例的一般性或问题的独立性。为了解决这四个问题，我们分别介绍了所谓的重缩放函数。然后将它们集成为一个功能，以便于实施。使用该积分函数，用于求解Euler/Navier-Stokes方程的四阶格式可以在光滑区域中以其优化的线性形式工作，并且在不连续处表现为非线性，以确保基本上无振荡的解。数值算例表明，它能够分辨从声学到激波的波，从亚音速到高超音速的流，以及从层流到湍流的流型。
A fourth-order, symmetric, weighted essentially non-oscillatory scheme is proposed with improved linear and nonlinear properties. In order to improve its linear property as the dispersion and dissipation relations, an optimization method is developed by solving a set of equations proposed in the paper. Using this approach, optimization objectives carefully chosen are realized, through which improved linear performance is attained while maintaining stability in the shock-reflection problem. Implementation of a nonlinear scheme is another important issue which usually affects its practical performance in applications. Optimization in this regard is thought to make the scheme work in its linear form effectively and problem-independently when the flow varies relatively smoothly. To fulfill the objectives, four topics were investigated. The first is a new hybrid indicator of smoothness which is based on the concept of total variation, through which good resolution is obtained for resolving structures with short wavelength. The second is a modification specially designed for the most downwind indicator to avoid numerical oscillations. The third is a new transition algorithm to make the scheme work between its linear and nonlinear states by a variable. The new algorithm is so designed as to avoid misjudgment of smooth and oscillatory flow field. The fourth one regards case generality or problem-independence. To address four issues, the so-called rescale functions are presented separately. They are then integrated as one function for easy implementation. Using this integrated function, a fourth-order scheme for solving the Euler/Navier-Stokes equations can work in its optimized linear form in the smooth region and behave nonlinearly at discontinuities to ensure essentially oscillation-free solutions. Numerical examples manifest its capabilities to resolve waves from acoustics to shocks, flows from subsonic to hypersonic speed, and flow patterns from laminar to turbulent.