Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields
Sensitivity Parameter-Independent Characteristic-Wise Well-Balanced Finite Volume WENO Scheme for the Euler Equations Under Gravitational Fields
复制标题
引力场下欧拉方程的灵敏度参数无关特性良好平衡有限体积WENO方案
DOI：
10.1007/s10915-021-01562-4
复制
发表时间：
2021
影响因子：
2.5
通讯作者：
Wai-Sun Don
中科院分区：
文献类型：
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作者：
Peng Li;Bao-Shan Wang;Wai-Sun Don
具有引力源项（PDE）的欧拉方程允许流体静力平衡状态，其中源项精确地平衡通量梯度。当用数值方法求解偏微分方程时，平衡点的精确保持性是非常理想的。Li和Xing（J Comput Phys 316：145-163，2016）针对等温平衡和多方平衡的情况提出了高阶平衡的特征量有限体积加权基本无振荡（FV-WENO）格式。与所声称的相反，该计划并不平衡。问题的根源在于韦诺多项式重建过程（韦诺算子）的非线性权重中的非零灵敏度参数的不稳定影响。在理论上证明了该方法的有效性，并在粗网格分辨率和长时间的偏微分方程数值模拟中得到了验证。在这项研究中，两个简单而有效的数值技术来自韦诺算子的乘性不变（MI）属性被调用来纠正敏感性参数的依赖性产生一个正确的证明的敏感性参数无关（特征明智）以及平衡的FV-WENO计划。的（非）以及平衡的性质的计划证明了几个一维和二维的基准稳态问题和一个小扰动的稳态问题。此外，还对引力场作用下的一维Sod问题进行了数值模拟，显示了该格式在捕捉激波、接触间断和稀疏波等方面的性能。计算结果还表明，采用三阶Runge-Kutta时间推进格式的数值格式，在不人为增加Lax-Friedrichs数值通量中的数值耗散的情况下，CFL数应小于0.5，以抑制激波处的Gibbs振荡。
Euler equations with a gravitational source term (PDEs) admit a hydrostatic equilibrium state where the source term exactly balances the flux gradient. The property of exact preservation of the equilibria is highly desirable when the PDEs are numerically solved. Li and Xing (J Comput Phys 316:145–163, 2016) proposed a high-order well-balanced characteristic-wise finite volume weighted essentially non-oscillatory (FV-WENO) scheme for the cases of isothermal equilibrium and polytropic equilibrium. On the contrary to what was claimed, the scheme is not well-balanced. The root of the problem is the precarious effects of a non-zero sensitivity parameter in the nonlinear weights of the WENO polynomial reconstruction procedure (WENO operator). The effects are identified in the theoretical proof for the well-balanced scheme and verified numerically on a coarse mesh resolution and a long time simulation of the PDEs. In this study, two simple yet effective numerical techniques derived from the multiplicative-invariance (MI) property of a WENO operator are invoked to rectify the sensitivity parameter’s dependency yielding a correct proof for the sensitivity parameter-independent (characteristic-wise) well-balanced FV-WENO scheme. The (non-)well-balanced nature of the schemes is demonstrated with several one- and two-dimensional benchmark steady state problems and a small perturbation over the steady state problems. Moreover, the one-dimensional Sod problem under the gravitational field is also simulated for showing the performance of the well-balanced FV-WENO scheme in capturing shock, contact discontinuity, and rarefaction wave in an essentially non-oscillatory nature. It also indicates that the numerical scheme with the third-order Runge–Kutta time-stepping scheme should take the CFL number less than 0.5 to mitigate the Gibbs oscillations at the shock without increasing the numerical dissipation artificially in the Lax–Friedrichs numerical flux.