Abstract: To accurately capture complex flow structures, high-order numerical algorithms for solving the Navier-Stokes equations have been extensively studied in recent years. The flux reconstruction (FR) scheme on unstructured meshes is a high-order framework that is easy to implement and achieves high accuracy while maintaining a compact stencil. The Kelvin-Helmholtz (K-H) instability is a typical shear-driven interfacial instability and is widely used as a simplified model for evaluating the resolution characteristics of numerical methods. The convergence of the FR scheme is first verified using the two-dimensional isentropic vortex problem. Then, compressible K-H instability is simulated at various resolutions and Reynolds numbers. By comparing statistical quantities (such as mean kinetic energy and mathematical entropy) and flow snapshots under different conditions, the results show that high-order schemes offer higher accuracy and numerical resolution. For the same number of degrees of freedom, increasing the polynomial order yields higher resolution than mesh refinement, though with reduced robustness. As the Reynolds number increases, the flow exhibits more pronounced multiscale features. Owing to their low numerical dissipation, high-order schemes can capture the complex multiscale structures that appear in shear-flow instability problems.