A subcell limiting based on high-order compact nonuniform nonlinear weighted schemes for high-order CPR method (Q6620253)

This study addresses the challenge of accurately capturing shocks and discontinuities in high-order numerical simulations of hyperbolic conservation laws, a crucial aspect of computational fluid dynamics (CFD) and related fields. Traditional high-order methods, such as the correction procedure via reconstruction (CPR) method, are prone to spurious oscillations when applied to problems with shocks or strong gradients. This paper proposes a hybrid approach, integrating the CPR method with a subcell-based compact nonuniform nonlinear weighted (CNNW) scheme, to balance the need for high resolution in smooth regions with robustness in handling discontinuities.\N\NThe authors first review the principles of the CPR method and the CNNW scheme, both of which rely on high-order polynomial interpolation and flux evaluation. The CPR method operates on structured and unstructured meshes and uses flux correction functions, while the CNNW scheme introduces nonlinear interpolation and compact flux operators to improve robustness. By detecting troubled cells -- regions prone to oscillations due to discontinuities -- the proposed hybrid scheme applies CNNW-based limiting selectively, thereby enhancing stability without sacrificing resolution.\N\NTo demonstrate the effectiveness of their approach, the authors apply their hybrid scheme, denoted as CPR-CNNW, to a series of benchmark problems: the one-dimensional Shu-Osher problem, the two-dimensional Riemann problem, and the shock-vortex interaction problem. These numerical experiments highlight the scheme's ability to achieve high resolution in smooth regions while accurately capturing shocks. The results show that increasing the number of troubled cells slightly reduces oscillations without compromising the scheme's overall resolution. Furthermore, the scheme performs consistently across different test cases, supporting its adaptability to complex flow conditions.\N\NThe significance of this research lies in its contribution to the development of high-order numerical methods that are both robust and accurate for a wide range of CFD applications. By combining the strengths of the CPR and CNNW methods, the proposed approach addresses a critical limitation of existing high-order schemes, paving the way for more reliable simulations in fields such as aeroacoustics, turbulence modelling, and shock-dominated flows. The modularity of the CPR-CNNW framework also suggests its potential applicability to other finite element methods, further broadening its impact. This work exemplifies a successful integration of theoretical innovation and practical applicability, advancing the state of high-order CFD methods.\N\NFor the entire collection see [Zbl 1515.65005].