Developing accurate, efficient and robust shock-capturing schemes on non-uniform grids remains challenging particularly when facing strong non-uniformity. Thus, we extend the unified normalised-variable diagram (UND) initially proposed by Deng (2023) [30] on uniform grids to non-uniform grids, and propose essentially non-oscillatory (ENO) and high-resolution regions in non-uniform grid UND. Based on the proposed UND, we formulate a high-resolution shock- capturing scheme termed ROUND (Reconstruction Operators on Unified-Normalise-variable Diagram) on non-uniform meshes. Unlike classic WENO (Weighted Essentially Non-Oscillatory) schemes applied to non-uniform grids, the ROUND scheme avoids the expensive calculation of smoothness indicators. The ROUND scheme is first applied to solve the scalar convection equation. The results reveal that the ROUND scheme significantly improves the numerical resolution and preserves the structure of the passively convected scalar compared to the TVD (Total Variation Diminishing) limiters. For capturing discontinuous solutions, the proposed ROUND scheme on non-uniform meshes surpasses the performance of the 5th-order WENO-JS scheme. The ROUND scheme is then integrated into discontinuous Galerkin (DG) with the FV subcell limiting method to enhance the numerical resolution at the subcell level while adhering to the discrete conservation law. The compactness and simplicity of the ROUND scheme on non-uniform grids are compatible with the DG method, known for its features such as compactness and flexibility of hp-refinement. The resulting DG method, utilising finite volume ROUND subcell limiting, is denoted as the DG/FV-ROUND scheme. To assess the accuracy and robustness of the DG/FV-ROUND scheme, we simulate high-speed compressible flows characterized by strong shock waves and small-scale flow structures. Comparative studies show the improved numerical resolution achieved by DG/FVROUND. Thus, this research demonstrates the efficacy and robustness of the ROUND scheme on non-uniform grids and affirms that incorporating high-resolution ROUND as subcell shock- capturing schemes can enhance the resolution of DG/FV methods.
