高精度有限差分格式的色散优化及耗散控制

The dispersion optimization and dissipation adjustment for high-order finite difference schemes

本文综述了我们在高精度有限差分格式的色散优化和耗散控制方面的研究进展。首先,我们提出了半离散有限差分格式色散和耗散相互独立的充分条件,实现了差分格式色散和耗散特性的独立调节。在此基础上,提出了色散最小、耗散可控的高精度差分格式,称为MDCD格式。MDCD格式已经得到了广泛应用,取得了很好的计算效果,但其主要缺点是耗散的调节依赖于经验。为了解决这一问题,我们进一步提出耗散的自适应调节方法。具有自适应耗散特性的高精度有限差分格式的基本特征是,差分格式的耗散能够随解的局部尺度自适应调节。为了构造这类格式,我们提出了一种新型的尺度识别器,它能够以等效无量纲波数的形式来定量衡量数值解的局部长度尺度。在此基础上,设计差分格式耗散参数与尺度识别器得到的等效无量纲波数之间的关系,从而构造了一类色散最小、耗散自适应的差分格式,称为MDAD格式。为了计算含有间断的问题,同时保持在光滑区的良好耗散特性,我们利用尺度识别器对一种经典的激波探测器进行改进,提出了一种新的激波探测器,并将自适应耗散格式与对应的WENO格式相混合,得到自适应耗散混合格式。近似色散关系显示该混合格式兼具高分辨率和鲁棒性。多个含间断流场的标准算例测试结果显示,自适应耗散混合格式具有良好的分辨率和激波捕捉能力。

In this paper,our recent work on the dispersion optimization and dissipation adjustment for high-order finite difference schemes is reviewed.For compressible flows with a broad range of length scales,good dispersion and dissipation properties are crucial for numerical schemes to realize high-fidelity simulations.It has been recognized that the minimization of dispersion error is an effective measure to improve the resolution of schemes,while the dissipation should be adjusted according to the local scale of the solution.We found a sufficient condition for a semi-discretized finite difference scheme to have independent dispersion and dissipation properties.Based on this finding,the high-order finite difference scheme with minimized dispersion and controllable dissipation(MDCD)was proposed.The MDCD scheme has been widely used and achieved good performance.However,one drawback is that the dissipation is controlled empirically.To realize the automatic adjustment of the dissipation according to the local scale of the solution,we devised a scale sensor to quantify the length scale of the numerical solution which can be calibrated to give the effective scaled wavenumbers.Then the dispersion-dissipation condition is used to construct the relationship between the dissipation parameter and the effective scaled wavenumber.A class of finite difference schemes with adaptive dissipation is thus obtained.To achieve the shock-capturing capability while maintaining the superior spectral properties in the smooth regions,we make an improvement to a classical shock detector by incorporating the proposed scale sensor,and construct a new hybrid scheme which consists of the adaptive dissipation scheme and the corresponding weighted essentially non-oscillation(WENO)scheme.The approximate dispersion relation(ADR)shows that the new hybrid scheme is accurate and robust.Several benchmark test problems with broadband length scales as well as discontinuities are presented to validate the high-resolution and the good shock-capturing capability of the proposed scheme.