Compact Finite Difference Method to solve Differential Equations

Supervisor : Dr. Harshita Madduri

A higher order numerical approach for solving 1-D parabolic PDEs, such as the Diffusion and Advection-Diffusion-Reaction (ADR) equations, is developed and studied in this work. The proposed approach uses the Crank–Nicolson traditional finite difference method (FDM) for time discretization and a fourth-order compact finite difference method (CFDM) for spatial discretization. The accuracy and efficiency limits of the explicit, implicit, and Crank-Nicolson finite difference methods (FDM) are highlighted in this we use operator-based and Taylor series techniques, a compact study is created to overcome these issues, producing a solution that uses fewer grid points while achieving higher accuracy. The stability of the scheme is analyzed using Von Neumann ( Fourier analysis). A comparative study of CFDM is conducted through Python implementation on a variety of problems involving diffusion, advection, and reaction with constant coefficients. The results show that CFDM yields more accurate and efficient solutions than traditional methods. The work concludes by outlining future directions, particularly the extension of the scheme to PDEs with variable coefficients, for which initial developments are underway.