Derive the finite difference scheme for the numerical solution of the first-order diffusion equation IBVP using the Crank-Nicolson method

Abstract

In the article, a differential scheme is created for the the first-order diffusion equation using the Crank-Nicolson method. The stability of the differential scheme was checked using the Neumann method. To solve the problem numerically, stability intervals were found using the Neman method. This work presents an analysis of the stability of the Crank-Nicolson scheme for the two-dimensional diffusion equation using Von Neumann stability analysis. The Crank-Nicolson scheme is a widely used numerical method for solving partial differential equations that combines the explicit and implicit schemes. The stability analysis is an important factor to consider when choosing a numerical method for solving partial differential equations, as numerical instability can cause inaccurate solutions. We show that the Crank-Nicolson scheme is unconditionally stable, meaning that it can be used for a wide range of parameters without being affected by numerical instability. Overall, the analysis and implementation presented in this work provide a framework for designing and analyzing numerical methods for solving partial differential equations using the Crank-Nicolson scheme. The stability analysis is crucial for ensuring the accuracy and reliability of numerical solutions of partial differential equations.