Abstract Based on the fifth-order scheme in our previous work (Deng et. al (2019) [28]), a new framework of constructing very high order discontinuity-capturing schemes is proposed for finite volume method. These schemes, so-called P n T m − BVD (polynomial of n-degree and THINC function of m-level reconstruction based on BVD algorithm), are designed by employing high-order upwind-biased interpolations and THINC (Tangent of Hyperbola for INterface Capturing) functions with adaptive steepness as the reconstruction candidates. The final reconstruction function in each cell is determined with a multi-stage BVD (Boundary Variation Diminishing) algorithm so as to effectively control numerical oscillation and dissipation. We devise the new schemes up to eleventh order in an efficient way by directly increasing the order of the underlying upwind scheme using high order polynomials. The analysis of the spectral property and accuracy tests show that the new reconstruction strategy well preserves the low-dissipation property of the underlying upwind schemes with high-order polynomials for smooth solution over all wave numbers and realizes n + 1 order convergence rate. The performance of new schemes is examined through widely used benchmark tests, which demonstrate that the proposed schemes are capable of simultaneously resolving small-scale flow features with high resolution and capturing discontinuities with low dissipation. With outperforming results and simplicity in algorithm, the new reconstruction strategy shows great potential as an alternative numerical framework for computing nonlinear hyperbolic conservation laws that have discontinuous and smooth solutions of different scales.

中文翻译：

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使用逆风偏置插值和边界变化减少算法构建高阶不连续捕获方案

摘要 基于我们之前工作中的五阶方案（Deng et. al (2019) [28]），提出了一种构建非常高阶不连续捕获方案的新框架，用于有限体积法。这些方案，所谓的 P n T m − BVD（n 阶多项式和基于 BVD 算法的 m 级重建的 THINC 函数），是通过采用高阶逆风偏置插值和 THINC（双曲线正切）设计的接口捕获）以自适应陡度作为重建候选函数。每个单元格中的最终重建函数采用多级BVD（Boundary Variation Diminishing）算法确定，以有效控制数值振荡和耗散。我们通过使用高阶多项式直接增加基础逆风方案的阶数，以有效的方式设计了高达 11 阶的新方案。频谱特性分析和精度测试表明，新的重构策略很好地保留了底层逆风方案的低耗散特性，具有高阶多项式，可平滑求解所有波数，并实现了 n+1 阶收敛速度。通过广泛使用的基准测试来检查新方案的性能，这表明所提出的方案能够同时解决高分辨率的小尺度流动特征和捕获低耗散的不连续性。凭借出色的结果和算法的简单性，