High order Numerical Methods for Solving differential equations in Engineering Applications

Abstract

High-order numerical methods have profoundly transformed the field of computational engineering by enabling the precise and efficient solution of differential equations that describe complex physical and mechanical phenomena. These methods—encompassing spectral techniques, high-order finite element formulations, and advanced Runge–Kutta time-integration schemes—provide significantly greater accuracy, faster convergence rates, and improved numerical stability compared to conventional low-order approaches. Their superior performance is especially evident in problems that demand high fidelity, such as crack propagation in solid mechanics, multi-dimensional heat and mass transfer, aerodynamic flow simulations, and vibration analysis of complex structures.