A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order
A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order
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DOI：
10.1016/j.jcp.2023.112052
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发表时间：
2023-03
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影响因子：
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通讯作者：
X. Deng
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文献类型：
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作者：
X. Deng
由于二阶插值格式比高阶插值格式具有更好的鲁棒性和算法简单性，三胞紧凑型网格中对流项的离散化在工程计算流体力学（CFD）中得到了广泛的应用。然而，根据Godnunov定理，在紧凑的模板中设计精确且无振荡的离散化方案从来都不是小事。几十年来，人们提出了各种各样的方案，以提高精度超过二阶，而不产生非物理的数值振荡。代表性的方案是基于多项式的非线性插值，如TVD（总变差递减），NVD（归一化变量图）和三阶韦诺（加权基本无振荡）;和非基于多项式的插值，包括RBF（径向基函数），LDLR（局部双对数重建）和双曲线的切线界面捕捉（THINC）。在半离散有限体积法中，现有的重建方案在一定程度上存在不同的问题。例如，现有的TVD方案倾向于引起拖尾、削波和平方效应。大多数韦诺格式和不相容RBF格式都不是尺度不变的。特别地，改进的WENO-Z格式既不是尺度不变的，也不是网格尺寸无关的。计算昂贵的非多项式插值通常具有奇异性，并依赖于可调参数。因此，这项工作审查重建方案通过重新制定或计算重建的边界值在规范化变量空间中的重建方案的特性可以直接读取。本文通过对现有重建方案的精度分析和归纳，提出了一种统一的归一化变量图（UND），它可以用图形的方式给出重建方案的以下性质：1）二阶和三阶精度条件，2）TVD和CBC（对流有界性准则）约束，3）ENO条件，4）高分辨率（HR）属性，5）以及数值耗散和反耗散误差.通过对已有格式的改进，首次证明了所提出的UND格式框架，并引入了一个保形TVD限制器和一个尺度-不变韦诺格式然后，通过定义归一化重构算子，提出了一种全新的重构方案ROUND（Reconstruction Operator on Unified Normalised-variable Diagram）。ROUND方案是通过直接配置UND上具有良好性质的归一化重建算子来设计的。这项工作提出了一个耗散的ROUND计划，其中归一化重建算子完全在耗散区域的UND，和一个低耗散的ROUND计划，其中严格调整的反耗散错误的介绍。不同配方的ROUND计划，以显示设计方案的灵活性UND框架内。通过基准测试对所提出的耗散和低耗散ROUND格式进行了评估，结果表明ROUND格式在精度和分辨率方面具有上级性能。特别是，在某些情况下，低耗散的ROUND格式获得可比的，甚至更好的分辨率比经典的五阶韦诺。将ROUND方法应用于有限差分-浸没边界法（FDM-IBM）框架的初步结果进一步证明了这一工作的重要性。
Discretization of the convection term in a three-cell compact stencil has been widely used in Computational Fluid Dynamics (CFD) of engineering because the second-order interpolation scheme is considered to be more robust and algorithmically simple than very high-order schemes. However, according to Godnunov's theorem, it has never been trivial to design an accurate and oscillation-free discretization scheme in a compact stencil. Over decades, various kinds of schemes have been proposed to increase accuracy beyond second-order without producing non-physical numerical oscillations. Representative schemes are polynomial-based non-linear interpolations such as TVD (Total Variation Diminishing), NVD (Normalized Variable Diagram) and third-order WENO (Weighted Essentially Non-oscillatory); and non-polynomial-based interpolations including RBF (Radial Basis Functions), LDLR (Local Double Logarithmic Reconstruction) and Tangent of Hyperbola for INterface Capturing (THINC). Within the semi-discretization finite volume method, existing reconstruction schemes suffer from different issues to some extent. For example, existing TVD schemes tend to cause smearing, clipping and squaring effects. Most WENO schemes and inconsistent RBF schemes are not scale-invariant. Especially, the improved WENO-Z scheme is neither scale-invariant nor mesh-size-independent. Computationally expensive non-polynomial-based interpolations usually have singularity and are dependent on tunable parameters. Thus, this work reviews reconstruction schemes by reformulating or calculating the reconstructed boundary value in the normalised-variable space where the characteristics of reconstruction schemes can be read directly. Through the accuracy analysis and the induction of existing schemes, this work proposes a Unified Normalised-variable Diagram (UND) where the following properties of a reconstruction scheme can be given diagrammatically: 1) the second-order and the third-order accuracy condition, 2) the TVD and CBC (Convection Boundedness Criterion) constrain, 3) the ENO condition, 4) the High Resolution (HR) property, 5) and the numerical dissipation and anti-dissipation error.The proposed UND framework is firstly demonstrated by the improvement of the existing schemes with a shape-preserving TVD limiter and a scale-invariant WENO scheme. Then, by defining the normalised reconstruction operator, a brand-new scheme named ROUND (Reconstruction Operator on Unified Normalised-variable Diagram) is proposed. The ROUND scheme is devised by directly configuring the normalised reconstruction operator of favourable properties on UND. This work presents a dissipative ROUND scheme in which normalised reconstruction operators are fully in the dissipative region of UND, and a low-dissipative ROUND scheme where rigorously adjusted anti-dissipation errors are introduced. Different formulations of ROUND schemes are presented to show the flexibility of designing schemes within the UND framework. The proposed dissipative and low-dissipative ROUND schemes are evaluated via benchmark tests which show the superior performance of ROUND on accuracy and resolution. Especially, in some cases, the low-dissipative ROUND schemes obtain comparable and even better resolution than classic fifth-order WENO. The significance of this work is further demonstrated by the preliminary results of applying ROUND on the FDM-IBM (Finite Difference Method and Immersed Boundary Method) framework.