Abstract In this paper, we present a new finite difference (FD) two stage boundary variation diminishing (BVD) P4T2 (fourth-degree polynomial (P4) and Tangent of Hyperbola for Interface Capturing (THINC) function with a two-level steepness (T2)) scheme for the compressible Euler equations. First, after splitting the flux vector into positive and negative flux vectors using an appropriate flux vector splitting methods, we use the P4 and THINC functions with a two-level steepness to obtain the upwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. Then, we adopt the corresponding downwind version of the P4 and THINC functions with a two-level steepness to obtain the downwind numerical fluxes at cell boundaries for both the positive and negative flux vectors. The final upwind numerical flux is determined for the positive/negative flux respectively through the two-stage BVD principle, which minimizes the jumps of the reconstructed values at the cell boundaries. The final scheme, called FD-P4T2-BVD, exhibits a better resolution, compared with WENO5 scheme, for small-scale structures, particularly the contact discontinuities in various one-dimensional and two-dimensional cases.

中文翻译：

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一种应用到欧拉方程的基于通量分裂的有限差分两阶段边界变化减少方案

摘要 在本文中，我们提出了一种新的有限差分 (FD) 两级边界变化递减 (BVD) P4T2（四阶多项式 (P4) 和双曲线切线，用于界面捕获 (THINC) 函数，具有两级陡度 (T2) )) 可压缩欧拉方程的方案。首先，在使用适当的通量向量分裂方法将通量向量分裂为正和负通量向量之后，我们使用具有两级陡度的 P4 和 THINC 函数来获得正和负通量的单元边界处的逆风数值通量向量。然后，我们采用具有两级陡度的 P4 和 THINC 函数的相应顺风版本，以获得正和负通量向量在单元边界处的顺风数值通量。通过两级 BVD 原理分别为正/负通量确定最终的逆风数值通量，该原理使单元边界处重构值的跳跃最小化。与 WENO5 方案相比，最终方案称为 FD-P4T2-BVD，对于小尺度结构，尤其是各种一维和二维情况下的接触不连续性，具有更好的分辨率。