High-order hybrid essentially non-oscillatory spectral difference method for hyperbolic conservation laws on unstructured curved quadrilateral grids

We propose a new hybrid limiting approach for the high-order spectral difference (SD) method for compressible flow computations. This approach preserves the key advantages of the SD method—such as compact stencils, flexibility on unstructured grids, and high-order accuracy—while enhancing its robustness near discontinuities. To achieve this, we extend the targeted essentially non-oscillatory (TENO) troubled-cell indicator (TCI) to unstructured grids composed of curved quadrilateral cells. The hybrid strategy applies the standard SD method in smooth regions and activates compact nonlinear limiters only in troubled cells identified by the TENO TCI. By embedding the TENO TCI directly into the linear weights of the compact nonlinear limiter, the contribution of low-order non-oscillatory reconstructions is set to zero in smooth regions, ensuring that high-order accuracy is maintained. As a result, parameter tuning in the nonlinear limiter primarily targets high-wavenumber regions and strong discontinuities. The proposed hybrid limiter demonstrates superior performance compared to compact weighted essentially non-oscillatory limiters, particularly in capturing small-scale turbulent structures near discontinuities. Extensive numerical experiments on both structured and unstructured grids validate the accuracy, robustness, and computational efficiency of the proposed method.