Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes
Development and stability analysis of the inverse Lax−Wendroff boundary treatment for central compact schemes
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DOI：
10.1051/m2an/2014024
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
François Vilar;Chi-Wang Shu
中科院分区：
文献类型：
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作者：
François Vilar;Chi-Wang Shu
在本文中，我们推广了所谓的逆Lax-Wendroff边界处理[S. Tan和C.- W. Shu，J. Comput. Phys.229（2010）8144-8166]的中心紧致格式[X. Liu，S. Zhang，H. Zhang和C.- W. Shu，J. Comput. 248（2013）235-256]。流出边界采用经典外推法处理，并对所得格式进行了稳定性分析。为了保证所考虑的格式的稳定性，G-K-S理论[B. Gustafsson，H. O. Kreiss和A.孙德斯特伦，数学计算。26（1972）649-686]，首先在半离散情况下，然后在具有三阶TVD Runge-Kutta时间离散的全离散情况下使用。然后，由于G-K-S理论的高代数复杂性，通过可视化离散化算子的特征谱来分析稳定性。我们在本文中表明，这两种不同的方法得到的结果是完全一致的。我们还说明了所提出的计划在简单的测试问题的高精度。
In this paper, we generalize the so-called inverse Lax−Wendroff boundary treatment [S. Tan and C.-W. Shu, J. Comput. Phys. 229 (2010) 8144–8166] for the inflow boundary of a linear hyperbolic problem discretized by the recently introduced central compact schemes [X. Liu, S. Zhang, H. Zhang and C.-W. Shu, J. Comput. Phys. 248 (2013) 235–256]. The outflow boundary is treated by the classical extrapolation and a stability analysis for the resulting scheme is provided. To ensure the stability of the considered schemes provided with the chosen boundaries, the G-K-S theory [B. Gustafsson, H.-O. Kreiss and A. Sundstrom, Math. Comput. 26 (1972) 649–686] is used, first in the semidiscrete case then in the fully discrete case with the third-order TVD Runge−Kutta time discretization. Afterwards, due to the high algebraic complexity of the G-K-S theory, the stability is analyzed by visualizing the eigenspectrum of the discretized operators. We show in this paper that the results obtained with these two different approaches are perfectly consistent. We also illustrate the high accuracy of the presented schemes on simple test problems.