This paper presents an analytical and computational investigation of the Compact Finite Difference Method (CFDM) and its applications to solving linear and nonlinear partial differential equations (PDEs). The CFDM, characterized by high-order spatial accuracy and minimal stencil width, provides superior resolution compared to traditional explicit schemes. The study includes derivation of compact finite difference schemes for first, second, and fourth spatial derivatives, followed by stability and convergence analysis using von Neumann analysis and eigenvalue methods. Numerical validation is demonstrated on model PDEs such as the 1D Heat Equation, Advection Equation, and Korteweg-de Vries (KdV) equation. Results demonstrate that CFDM achieves sixth-order accuracy in space with enhanced spectral fidelity and computational efficiency.
Rakesh saini (Thu,) studied this question.