ADVANCED NUMERICAL METHODS FOR SOLVING NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN FLUID MECHANICS: APPLICATIONS IN AEROSPACE ENGINEERING

Abstract

The solution of nonlinear partial differential equations (PDEs) in fluid mechanics remains one of the most critical and challenging tasks in computational science, particularly within the domain of aerospace engineering. These equations, primarily derived from the Navier–Stokes framework, govern a wide range of complex phenomena including turbulence, compressible flows, boundary-layer separation, shock–boundary interactions, and hypersonic aerodynamics. Traditional numerical techniques such as finite difference and finite volume methods, while effective in linear or moderately nonlinear regimes, often suffer from limitations when extended to strongly nonlinear, multi-scale problems characterized by high Reynolds numbers and stiff temporal dynamics. In response, recent advances have introduced more robust and adaptive strategies, including spectral methods, high-order finite element formulations, lattice Boltzmann approaches, and machine learning–augmented solvers, all of which have demonstrated significant promise in enhancing accuracy, stability, and computational efficiency. This paper critically examines these advanced numerical methods, with a focus on their comparative performance in aerospace applications such as aerodynamic load prediction, flow stability in re-entry vehicles, shock-capturing in supersonic jets, and turbulence modelling in propulsion systems. Through an integrated perspective, the study highlights both the theoretical underpinnings and practical implementations of these approaches, while emphasizing the importance of high-performance computing and hybrid schemes for tackling real-world aerospace design challenges. The analysis contributes to bridging the gap between mathematical theory and engineering practice, providing a roadmap for future research in nonlinear PDE modelling within aerospace fluid mechanics.