Novel efficient, high-resolution and structure-preserving convection schemes for computational fluid dynamics

In this work, we present our recent efforts to develop high-resolution, scalar-structure-preserving schemes within a three-cell-based compact stencil. Our approach begins with unifying existing three-cell-based non-linear reconstruction schemes into a single framework through the definition of a Unified Normalised-Variable Diagram (UND). Using UND, the reconstruction operator can be designed directly in the normalised-variable space to satisfy desirable properties such as Total Variation Diminishing (TVD) and Essentially Non-Oscillatory (ENO). The resolution of the reconstruction operator is enhanced by introducing anti-diffusion errors to UND, while preserving the scalar structure through fine-tuning these errors. The resultant novel scheme is termed ROUND (Reconstruction Operator on Unified Normalised-Variable Diagram). The proposed ROUND schemes are evaluated using benchmark tests, demonstrating their superior performance in terms of accuracy and resolution. Notably, in some cases, the low-dissipative ROUND schemes achieve comparable or even better resolution than the classic fifth-order Weighted ENO (WENO). We further extend and implement the ROUND scheme on unstructured grids within the OpenFOAM framework. Compared to conventional second-order schemes in OpenFOAM, ROUND significantly reduces numerical errors at a similar computational cost. Additionally, ROUND offers improved structure-preserving properties over conventional schemes. The performance of ROUND schemes is further evaluated by simulating high-speed compressible single-phase and multiphase flows. We also demonstrate the efficacy of ROUND schemes by extending them to the Finite Difference Method Immersed Boundary Method (FDM-IBM) and Discontinuous Galerkin methods, thereby enhancing robustness and numerical resolution.