Complex nonlinear systems, governed by partial differential equations, are ubiquitous across science and engineering from computational fluid dynamics and plasma physics to phase-field modeling and biological pattern formation. While high-order numerical methods offer superior accuracy per degree of freedom, their practical deployment for such systems faces persistent challenges: maintaining provable stability and physical fidelity (e.g., entropy stability, positivity preservation, energy conservation/dissipation), ensuring computational efficiency on modern architectures, and handling complex geometries with unstructured meshes . This mini-symposium brings together recent advances in the development of high-order accurate numerical schemes that address these challenges through rigorous mathematical design and algorithmic innovation.