Abstract
The goal of this research is to develop and validate new computational approaches to enhance our understanding of fluid dynamics.To provide more accurate tools for industrial operations, meteorology, and aeronautical engineering, we address the limitations of existing methods.We employed third-order Runge-Kutta methods with Total Variation Diminishing (TVD) for temporal integration, fifth-order Weighted Essentially Non-Oscillatory (WENO) schemes for spatial discretization, as well as Finite Volume Methods (FVM) and Finite Element Methods (FEM) as advanced numerical techniques.The novelty of this work lies in integrating high-order WENO schemes with FEM and AMR for full compressible Navier-Stokes equations with moving boundaries and temperaturedependent heat conductivity, which has not been previously addressed in the literature.Additionally, high-performance computing methods, moving mesh approaches, and Adaptive Mesh Refinement (AMR) were utilized.The results demonstrate significant improvements in both the efficiency and accuracy of the simulations.Specifically, compared to traditional second-order methods, the fifth-order WENO schemes reduced errors by a factor of four.Furthermore, it was shown that the new schemes enhanced the accuracy of capturing discontinuities and fine-scale structures, maintaining a variation of less than 1% from analytical solutions, while reducing computational complexity by up to 30% and processing time by approximately 25%.These findings suggest that the proposed WENO schemes offer multiple valuable advantages for high-precision applications in hydrodynamics and aeronautical engineering, which solve hyperbolic conservation laws.