Comparative Study of Crank-Nicolson and Modified Crank-Nicolson Numerical methods to solve linear Partial Differential Equations

Objectives: This paper aims to address the limitations of the Crank-Nicolson Finite Difference method and propose an improved version called the modified Crank-Nicolson method. Methods: Utilized implicit discretization in time and space, with parameters k = 0.001, h = 0.1, and γ = 0.1. Conducted extensive testing on various partial differential equations. Findings: Results, displayed in Table 1, showcase the method's stability and accuracy. Comparative analysis in Table 2 demonstrates the Modified Crank-Nicolson method consistently outperforming the traditional approach, reaffirming its superiority in accuracy. Novelty: The modified Crank-Nicolson method offers a significant enhancement to the traditional Crank-Nicolson finite difference method, making it a valuable tool for effectively solving partial differential equations. Keywords: Crank­Nicolson Method, Modified Crank­Nicolson Method, Finite Difference, Partial Differential Equations, Parabolic Equations, Python Software