A Superlative Study on the Numerical Solutions of Partial Differential Equations Using Finite Difference, Finite Element, and Finite Volume Methods

Partial Differential Equations (PDEs) play a crucial role in modeling physical and engineering
phenomena, including heat transfer, wave propagation, and fluid dynamics. Due to the complexity of
obtaining analytical solutions, numerical methods are widely employed for solving PDEs. This study
presents a comparative analysis of three numerical techniques: the Finite Difference Method (FDM),
the Finite Element Method (FEM), and the Finite Volume Method (FVM). The research evaluates the
stability, accuracy, efficiency, and convergence of these methods by applying them to benchmark
problems, such as the heat equation and the wave equation. The methodology involves implementing
each technique for solving selected PDEs, followed by an assessment of their performance based on
computational complexity and error analysis. The results highlight the strengths and limitations of
each method: FDM is computationally efficient for simple geometries but requires strict stability
conditions, FEM offers flexibility in handling complex geometries and boundary conditions, and
FVM ensures local conservation properties, making it particularly useful in fluid dynamics
applications. The study concludes that the choice of numerical method depends on the specific
characteristics of the problem, including domain geometry, boundary conditions, and computational
constraints. This comparative analysis provides valuable insights into selecting appropriate
numerical techniques for solving PDEs in practical applications.