A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order

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.