The Navier-Stokes (NS) equations are fundamental mathematical models with a wide range of applications in computational math and various engineering fields. The major objective of this project is to explore high-order accurate structure-preserving methods for various NS equations, including compressible, incompressible, and compressible flow with incompressible limit. The outcomes have potential applications in aeronautics and astronautics, petroleum industries for enhancing oil recovery, and possibly extended to benefit computational materials science. In addition, this project will provide valuable research opportunities for graduate and/or undergraduate students in computational mathematics. Designing high-order methods for NS equations that preserve fundamental principles such as conservation, bounds, and energy law, while ensuring efficiency for large-scale simulations, presents significant challenges. The current state of high-order accurate structure-preserving numerical methods for NS equations is still far from being practically satisfactory. The PI will explore high-order structure-preserving algorithms for various NS equations, emphasizing efficiency and robustness in simulating real-world problems. For compressible NS, a novel approach combining large-scale non-smooth optimization with discontinuous Galerkin methods will be applied to construct invariant-domain-preserving schemes. This methodology is extensible to other methods, such as finite volume and finite